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Hypothesis Testing February 25, 2020 Data Science CSCI 1951A Brown - PowerPoint PPT Presentation

Dec 11, 2022 •270 likes •1.37k views

Hypothesis Testing February 25, 2020 Data Science CSCI 1951A Brown University Instructor: Ellie Pavlick HTAs: Josh Levin, Diane Mutako, Sol Zitter 1 Announcements .? 2 Today Two quick preliminaries: LoLN and CLT Follow up

  1. Clicker Question! Every year, I compute the mean grade in my class. I never change the material or my methods for evaluating because, lazy. Over the 439 years that I Central Limit Theorem: repeated have been teaching this class, this has resulted in the below distribution. measures of mean will be normally 40 30 distributed, doesn’ t assume the 20 population over which you are taking 10 0 the mean is normally distributed. 10 20 30 40 50 60 70 80 90 100 Which of these is mostly like the typical distribution on any given year? 40 40 (c) can’t say, (a) (b) 30 30 could be 20 20 10 10 either 0 0 10 20 30 40 50 60 70 80 90100 10 20 30 40 50 60 70 80 90100 15

  2. Today • Two quick preliminaries: LoLN and CLT • Follow up from last time • Common tests: chi-squared test, z-test/t-tests 16

  3. Hypothesis Testing in General • Null hypothesis (H 0 ) — the “nothing to see here” assumption • Alternative hypothesis (H a )—the thing you know will lead to an explosive headline and are really hoping is true but you are a good scientist, so you will look to the data to confirm 17

  4. Hypothesis Testing in General • Null hypothesis (H 0 ) — the “nothing to see here” assumption • Alternative hypothesis (H a )—the thing you know will lead to an explosive headline and are really hoping is true but you are a good scientist, so you will look to the data to confirm 18

  5. Hypothesis Testing in General • Null hypothesis (H 0 ) — the “nothing to see here” assumption • Alternative hypothesis (H a )—the thing you know will lead to an explosive headline and are really hoping is true but you are a good scientist, so you will look to the data to confirm 19

  6. Hypothesis Testing in General Thing you can model • Null hypothesis (H 0 ) — the “nothing to see here” assumption • Alternative hypothesis (H a )—the thing you know will lead to an explosive headline and are really hoping is true but you are a good scientist, so you will look to the data to confirm 20

  7. Hypothesis Testing in General Thing you can model • Null hypothesis (H 0 ) — the “nothing to see here” assumption • Alternative hypothesis (H a )—the thing you know will lead to an explosive headline and are really hoping is true but you are a good scientist, so you will look to the data to confirm Matters for how you compute p-values…more soon 21

  8. Hypothesis Testing in General • Assume the null hypothesis is true—i.e., don’t deviate from status quo without good reason :) • If there is enough evidence to suggest that H 0 is highly unlikely, then we can say we “reject the null hypothesis” • If there is not enough evidence, we “fail to reject it” • We don’t “accept” or “prove” H 0 or H a 22

  9. Hypothesis Testing in General • Determine an appropriate test statistic, given your null hypothesis • Come up with a theoretical distribution of that test statistic (often, this work has been done for you) • Compute the likelihood of your test statistic, given that theoretical distribution 23

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  12. <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> Hypothesis Testing Observation/Sample “I swear literally like 80% of the answers are just (b)” h Ω , F, P i c d b d b a d c 4 {b, not b} p(B) = 0.8 b c b d cdf 1.00 0.93 1 0.73 0.75 0.44 0.5 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 26

  13. <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> Hypothesis Testing Test Statistic “I swear literally like 80% of the answers are just (b)” h Ω , F, P i c d b d b a d c 4 {b, not b} p(B) = 0.8 b c b d cdf 1.00 0.93 1 0.73 0.75 0.44 0.5 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 27

  14. <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> Hypothesis Testing Theoretical Distribution “I swear literally like 80% of the answers are just (b)” h Ω , F, P i c d b d b a d c 4 {b, not b} p(B) = 0.8 b c b d cdf 1.00 0.93 1 0.73 0.75 0.44 0.5 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 28

  15. <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> Hypothesis Testing “I swear literally like 80% of the answers are just (b)” h Ω , F, P i c d b d b a d c 4 {b, not b} p(B) = 0.8 b c b d cdf 1.00 0.93 1 p-value 0.73 0.75 P(#(b)s ≤ 4) 0.44 0.5 = 0.0006 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 29

  16. Today • Two quick preliminaries: LoLN and CLT • Follow up from last time • Common tests: chi-squared test, z-test/t-tests 30

  17. Some tests you are likely to use • t-test : difference of means; is the average value of some feature different between two populations • e.g. are men taller than women, are blue states more populated than red states, do CS work harder than other majors (::rolling_eyes::) • chi-squared test : difference in frequencies of a categorical variable; is the distribution of some feature uniform across groups • e.g. do neighborhoods differ in terms of music preferences features; do college majors differ in terms of sociodemographic features • correlation : do two variables tend to move together (usually best for exploratory analyses, rather than hypothesis testing) • null hypothesis: “there is no correlation between X and Y” 31

  18. Some tests you are likely to use • t-test : difference of means; is the average value of some feature different between two populations • e.g. are men taller than women, are blue states more populated than red states, do CS work harder than other majors (::rolling_eyes::) • chi-squared test : difference in frequencies of a categorical variable; is the distribution of some feature uniform across groups • e.g. do neighborhoods differ in terms of music preferences features; do college majors differ in terms of sociodemographic features 32

  19. Some tests you are likely to use • t-test : difference of means; is the average value of some feature different between two populations • e.g. are men taller than women, are blue states more populated than red states, do CS work harder than other majors (::rolling_eyes::) • chi-squared test : difference in frequencies of a categorical variable; is the distribution of some feature uniform across groups • e.g. do neighborhoods differ in terms of music preferences features; do college majors differ in terms of sociodemographic features 33

  20. Some tests you are likely to use • t-test: difference of means; is the average value of some feature different between two populations • e.g. are men taller than women, are blue states more populated than red states, do CS work harder than other majors (::rolling_eyes::) • chi-squared test : difference in frequencies of a categorical variable; is the distribution of some feature uniform across groups • e.g. do neighborhoods differ in terms of music preferences features; do college majors differ in terms of sociodemographic features 34

  21. Test for population means 35

  22. Test for population means Distribution of ages in the US Hypothesis: Mean age is 35. 36 http://www.censusscope.org/us/chart_age.html

  23. Test for population means Distribution of ages in the US Hypothesis: Mean age is 35. Thoughts about test statistics? 37 http://www.censusscope.org/us/chart_age.html

  24. <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> Test for population means Distribution of ages in the US z = ¯ x − µ 0 q σ 2 n Hypothesis: Mean age is 35. 38 http://www.censusscope.org/us/chart_age.html

  25. <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> Test for population means observed Distribution of ages in the US z = ¯ x − µ 0 q σ 2 n Hypothesis: Mean age is 35. 39 http://www.censusscope.org/us/chart_age.html

  26. <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> Test for population means Distribution of ages in the US z = ¯ x − µ 0 q σ 2 expected n Hypothesis: Mean age is 35. 40 http://www.censusscope.org/us/chart_age.html

  27. <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> Test for population means Distribution of ages in the US z = ¯ x − µ 0 q σ 2 n variation (comes from normal Hypothesis: distribution) Mean age is 35. 41 http://www.censusscope.org/us/chart_age.html

  28. Clicker Question! 42

  29. Clicker Question! Why can we use a normally-distributed test statistic to evaluate mean age of a population? a) Because ages are normally distributed b) Because the test statistic is a random variable c) Because of the law of large numbers d) Because of the central limit theorem e) The limit does not exist! 43

  30. Clicker Question! Why can we use a normally-distributed test statistic to evaluate mean age of a population? a) Because ages are normally distributed b) Because the test statistic is a random variable c) Because of the law of large numbers d) Because of the central limit theorem e) The limit does not exist! 44

  31. <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> Test for population means z = ¯ x − µ 0 q σ 2 n Distribution of mean ages given samples of US adults 0 45 http://www.censusscope.org/us/chart_age.html

  32. Standard Normal Distr. 0 z = distance from mean in std units 46

  33. Standard Normal Distr. 0 z = (observed - expected) / standard deviation 47

  34. Standard Normal Distr. observed test statistic 0 z = (observed - expected) / standard deviation 48

  35. Standard Normal Distr. cumulative density = p-value 0 z = (observed - expected) / standard deviation 49

  36. Detour! 𝛃 and determining “significance” 50

  37. <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> <latexit sha1_base64="EiaYKOL1VTSk+93M8x0Av3/NtkA=">ACBXicbVA9SwNBEN2LXzF+nVpqsRgEixDuRNAyKIidEcwH5ELY20ySJbt7x+6eEI40Nv4VGwtFbP0Pdv4bN5cUmvhg4PHeDPzwpgzbTzv28ktLa+sruXCxubW9s7u5eXUeJolCjEY9UMyQaOJNQM8xwaMYKiAg5NMLh1cRvPIDSLJL3ZhRDW5C+ZD1GibFSxz0MOJF9Dji4FdAnJXxdwlUcqEzsuEWv7GXAi8SfkSKaodpxv4JuRBMB0lBOtG75XmzaKVGUQ7jQpBoiAkdkj60LJVEgG6n2RdjfGyVLu5FypY0OFN/T6REaD0Soe0UxAz0vDcR/NaieldtFMm48SApNFvYRjE+FJLjLFDR5YQqpi9FdMBUYQaG1zBhuDPv7xI6qdl3yv7d2fFyuUsjw6QEfoBPnoHFXQDaqiGqLoET2jV/TmPDkvzrvzMW3NObOZfQHzucPOdGXGg=</latexit> Detour! 𝛃 and determining “significance” “I swear literally like 80% of the answers are just (b)” h Ω , F, P i c d b d b a d c 4 {b, not b} p(B) = 0.8 b c b d cdf 1.00 0.93 1 p-value 0.73 0.75 P(#(b)s ≤ 4) 0.44 0.5 = 0.0006 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 51

  38. Detour! 𝛃 and determining “assuming the null hypothesis “significance” were true, I would only expect to see a test statistic like this 0.006% of the time. How surprising! c d b d b a d c 4 b c b d cdf 1.00 0.93 1 p-value 0.73 0.75 P(#(b)s ≤ 4) 0.44 0.5 = 0.0006 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 52

  39. Detour! 𝛃 and determining “assuming the null hypothesis “significance” were true, I would only expect to see a test statistic like this 0.006% of the time. How surprising! c d b d b a d c 4 b c b d cdf 1.00 0.93 1 p-value 0.73 0.75 P(#(b)s ≤ 4) 0.44 0.5 = 0.0006 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 53

  40. Detour! 𝛃 and determining “assuming the null hypothesis “significance” were true, I would only expect to see a test statistic like this 0.006% of the time. How surprising! c d b d b a d c 4 b c b d cdf need to decide in 1.00 0.93 1 advance how willing p-value 0.73 0.75 you are to be P(#(b)s ≤ 4) 0.44 0.5 surprised ( 𝛃 ) = 0.0006 0.21 0.25 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 54

  41. Standard Normal Distr. significance level (set in advance) cumulative density = 𝛃 p-value 0 z = (observed - expected) / standard deviation 55

  42. Standard Normal Distr. Ha = test stat significance level is less than (set in advance) expected cumulative “one tailed” density = 𝛃 p-value 0 z = (observed - expected) / standard deviation 56

  43. assuming the null hypothesis is true, you will be still be “surprised” alpha Standard Normal Distr. % of the time Ha = test stat significance level is less than (set in advance) expected cumulative “one tailed” density = 𝛃 p-value 0 z = (observed - expected) / standard deviation 57

  44. assuming the null hypothesis is true, you will be still be “surprised” alpha Standard Normal Distr. % of the time Ha = test stat significance level is different (set in advance) than expected cumulative ??? “one tailed” density = 𝛃 p-value 0 z = (observed - expected) / standard deviation 58

  45. assuming the null hypothesis is true, you will be still be “surprised” alpha Standard Normal Distr. % of the time Ha = test stat significance level is different (set in advance) than expected cumulative 𝛃 /2 “two tailed” 𝛃 /2 density = p-value 0 z = (observed - expected) / standard deviation 59

  46. <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> Test for population means z = ¯ x − µ 0 q σ 2 n H 0 = mean age is 35 H a = mean age is not 35 𝛃 /2 𝛃 /2 0 60 http://www.censusscope.org/us/chart_age.html

  47. <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> <latexit sha1_base64="zqIv5EpLuHtNgnVynuUg6FmDC1o=">ACHnicbVDLSgMxFM34rPVdekmWAQ3lpmi6EYounFZwT6gU0smzbShSWZMmIN8yVu/BU3LhQRXOnfmLaz0NYDFw7n3Mu9wQxo0q7rczN7+wuLScW8mvrq1vbBa2tusqSiQmNRyxSDYDpAijgtQ01Yw0Y0kQDxhpBIOLkd+4I1LRSFzrYUzaHPUEDSlG2kqdwvEDPIN+KBE2foCkuU/hIfR50nFT46tbqU1mKtrj6KacGpGmadQdEvuGHCWeBkpgzVTuHT70Y4URozJBSLc+NdsgqSlmJM37iSIxwgPUIy1LBeJEtc34vRTuW6ULw0jaEhqO1d8TBnGlhjywnRzpvpr2RuJ/XivR4WnbUBEnmg8WRQmDOoIjrKCXSoJ1mxoCcKS2lsh7iMbh7aJ5m0I3vTLs6ReLnluybs6KlbOszhyYBfsgQPgRNQAZegCmoAg0fwDF7Bm/PkvDjvzsekdc7JZnbAHzhfP1Uso0g=</latexit> Test for population means z = ¯ x − µ 0 😠 q σ 2 n H 0 = mean age is 35 H a = mean age is not 35 𝛃 /2 𝛃 /2 0 61 http://www.censusscope.org/us/chart_age.html

  48. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Test for population means z = ¯ x − µ 0 s √ n H 0 = mean age is 35 standard normal H a = mean age is not 35 𝛃 /2 𝛃 /2 0 62 http://www.censusscope.org/us/chart_age.html

  49. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Test for population means z = ¯ x − µ 0 s √ n H 0 = mean age is 35 standard normal H a = mean age is not 35 students t 𝛃 /2 𝛃 /2 0 63 http://www.censusscope.org/us/chart_age.html

  50. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Test for population means z = ¯ x − µ 0 s √ n H 0 = mean age is 35 standard normal H a = mean age is not 35 closer to standard students t normal as n 𝛃 /2 𝛃 /2 increases 0 64 http://www.censusscope.org/us/chart_age.html

  51. Is there evidence to reject the null hypothesis? Null Hypothesis: The average grade is 85%. Grades 90 92 80 87 98 78 65

  52. Clicker Question! 66

  53. Clicker Question! What is the mean? Null Hypothesis: The average grade is 85%. Grades a) 85 90 92 b) 87 .5 80 c) 90 87 d) 92.5 98 78 67

  54. Clicker Question! What is the mean? Null Hypothesis: The average grade is 85%. Grades a) 85 90 92 b) 87 .5 80 c) 90 87 d) 92.5 98 78 68

  55. Clicker Question! What is the standard error? Null Hypothesis: The average grade is 85%. Grades a) √ 47 .25 90 92 b) √ 51.3 80 c) √ 56.7 87 d) √ 57 .25 98 78 69

  56. Clicker Question! What is the standard error? Null Hypothesis: The average grade is 85%. Grades a) √ 47 .25 90 92 b) √ 51.3 80 c) √ 56.7 87 d) √ 57 .25 98 78 70

  57. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Clicker Question! What is the test statistic? Null Hypothesis: The average grade is 85%. z = ¯ x − µ 0 Grades 90 s a) 0.82 92 √ n b) 0.85 80 87 c) 0.91 mean: 87.5 98 d) 0.95 s: 7.5 78 71

  58. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Clicker Question! What is the test statistic? Null Hypothesis: The average grade is 85%. z = ¯ x − µ 0 Grades 90 s a) 0.82 92 √ n b) 0.85 80 87 c) 0.91 mean: 87.5 98 d) 0.95 s: 7.5 78 72

  59. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Is there evidence to reject the null hypothesis? Null Hypothesis: The average grade is 85%. z = ¯ x − µ 0 Grades 90 s a) 0.82 92 √ n b) 0.85 80 87 c) 0.91 mean: 87.5 98 d) 0.95 s: 7.5 78 73

  60. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Is there evidence to reject the null hypothesis? Null Hypothesis: The average grade is 85%. z = ¯ x − µ 0 Grades 90 s a) 0.82 92 √ n b) 0.85 80 87 c) 0.91 mean: 87.5 98 d) 0.95 s: 7.5 78 74

  61. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Is there evidence to reject the null hypothesis? Null Hypothesis: The average grade is 85%. z = ¯ x − µ 0 Grades 90 s a) 0.82 92 √ n t t e b) 0.85 s t 80 _ 1 s a m p 87 c) 0.91 mean: 87.5 98 d) 0.95 s: 7.5 78 75

  62. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Test for difference in means z = ¯ x − µ 0 s √ n 0 76 http://www.censusscope.org/us/chart_age.html

  63. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Test for difference in means z = ¯ x − µ 0 s √ n mean 1 - mean 2 0 77 http://www.censusscope.org/us/chart_age.html

  64. <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> <latexit sha1_base64="QTywTK8kfHKaPNc+P42VqbwkTOA=">ACF3icbZDLSsNAFIYn9VbrLerSzWAR3BgSEXQjFN24rGAv0JQwmU7aoZNJnJmIdchbuPFV3LhQxK3ufBunbRba+sPAx3/O4cz5w5RqVz32yotLC4tr5RXK2vrG5tb9vZOUyaZwKSBE5aIdogkYZSThqKkXYqCIpDRlrh8HJcb90RIWnCb9QoJd0Y9TmNKEbKWIHtPMBz6EcCYe2HSOj7HB5BP84CN9dTWxqQt0Jpnud5YFdx50IzoNXQBUqgf2l9LcBYTrjBDUnY8N1VdjYSimJG84meSpAgPUZ90DHIUE9nVk7tyeGCcHowSYR5XcOL+ntAolnIUh6YzRmogZ2tj879aJ1PRWVdTnmaKcDxdFGUMqgSOQ4I9KghWbGQAYUHNXyEeIJOGMlFWTAje7Mnz0Dx2PNfxrk+qtYsijLYA/vgEHjgFNTAFaiDBsDgETyDV/BmPVkv1rv1MW0tWcXMLvgj6/MH9CegeA=</latexit> Test for difference in means z = ¯ x − µ 0 s √ n mean 1 - mean 2 zero 0 78 http://www.censusscope.org/us/chart_age.html

  65. Is there evidence to reject the null hypothesis? Null Hypothesis: The average grade is the same as last year. Alt. Hypothesis: This year’s grades are higher. 2019 2020 90 95 92 92 80 83 87 87 98 98 78 75 79

  66. Is there evidence to reject the null hypothesis? Null Hypothesis: The average grade is the same as last year. Alt. Hypothesis: This year’s grades are higher. 2019 2020 90 95 92 92 t t e s t _ i 80 83 n d 87 87 98 98 78 75 80

  67. Some tests you are likely to use • t-test: difference of means; is the average value of some feature different between two populations • e.g. are men taller than women, are blue states more populated than red states, do CS work harder than other majors (::rolling_eyes::) • chi-squared test: difference in frequencies of a categorical variable; is the distribution of some feature uniform across groups • e.g. do neighborhoods differ in terms of music preferences features; do college majors differ in terms of sociodemographic features 81

  68. Are the answers to my clicker questions random? c d b d b a d c b c b d 82

  69. Are the answers to my clicker questions random? 4 4 4 3 3 2 1 1 0 a b c d 83

  70. Are the answers to my clicker questions random? 4 4 4 3 3 2 1 1 0 a b c d Xi = count of answer i p(a) = p(b) = p(c) = p(d) = 0.25 84

  71. Are the answers to my clicker questions random? Is this distribution 4 4 4 significantly different than 3 what we would expect by 3 chance, assuming that in 2 fact all answers are 1 1 equally likely? 0 a b c d Xi = count of answer i p(a) = p(b) = p(c) = p(d) = 0.25 85

  72. Are the answers to my clicker questions random? 4 4 3 3 3 3 4 3 3 3 2.25 2 1.5 1 1 0.75 0 0 a b c d a b c d Observed Expected 86

  73. Are the answers to my clicker questions random? 4 4 3 3 3 3 4 3 3 3 2.25 2 1.5 1 1 0.75 0 0 a b c d a b c d Observed Expected Want to model the difference between these 87

  74. <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> Are the answers to my clicker questions random? ( X i − E ( X i )) 1 1 1 0.5 0 0 -0.5 -1 -1.5 -2 -2 a b c d 88

  75. Clicker Question! 89

  76. <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> Clicker Question! Should I use the total difference between observed and expected as my summary statistic? I.e. X ( X i − E ( X i )) i a) Yes! That sounds good. b) No! I have qualms… 90

  77. <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> <latexit sha1_base64="ur9WiI/2gQH7cMlO30XbibR+BW8=">AB/nicbVDLSgMxFL1TX7W+RsWVm2AR2oVlRgRdFkVwWcE+oB2GTJq2oZnMkGSEMhT8FTcuFHrd7jzb8y0s9DWA8k9nHMvuTlBzJnSjvNtFVZW19Y3ipulre2d3T17/6ClokQS2iQRj2QnwIpyJmhTM81pJ5YUhwGn7WB8k/ntRyoVi8SDnsTUC/FQsAEjWBvJt496Kgl9hiodc52h26xWq75dmrODGiZuDkpQ46Gb3/1+hFJQio04VipruvE2kux1IxwOi31EkVjTMZ4SLuGChxS5aWz9afo1Ch9NIikOUKjmfp7IsWhUpMwMJ0h1iO16GXif1430YMrL2UiTjQVZP7QIOFIRyjLAvWZpETziSGYSGZ2RWSEJSbaJFYyIbiLX14mrfOa69Tc+4ty/TqPowjHcAIVcOES6nAHDWgCgRSe4RXerCfrxXq3PuatBSufOYQ/sD5/AC2vk6o=</latexit> Clicker Question! Should I use the total difference between observed and expected as my summary statistic? I.e. X ( X i − E ( X i )) i a) Yes! That sounds good. b) No! I have qualms… 91

  78. <latexit sha1_base64="+hL14ZS6P1maznpjne569gjGrlQ=">ACAnicbVDLSgMxFM3UV62vUVfiJliEdmGZEUE3QlElxXsA9phyKSZNjTJDElGKENx46+4caGIW7/CnX9jp2Fth5I7uGce0nuCWJGlXacb6uwtLyulZcL21sbm3v2Lt7LRUlEpMmjlgkOwFShFBmpqRjqxJIgHjLSD0Xmtx+IVDQS93ocE4+jgaAhxUgbybcPeirhPoWVjrlO4E1Wq1V4CR3fLjs1Zwq4SNyclEGOhm9/9foRTjgRGjOkVNd1Yu2lSGqKGZmUeokiMcIjNCBdQwXiRHnpdIUJPDZKH4aRNEdoOFV/T6SIKzXmgenkSA/VvJeJ/3ndRIcXkpFnGgi8OyhMGFQRzDLA/apJFizsSEIS2r+CvEQSYS1Sa1kQnDnV14krdOa69Tcu7Ny/SqPowgOwRGoABecgzq4BQ3QBg8gmfwCt6sJ+vFerc+Zq0FK5/ZB39gf4A4liUfw=</latexit> <latexit sha1_base64="+hL14ZS6P1maznpjne569gjGrlQ=">ACAnicbVDLSgMxFM3UV62vUVfiJliEdmGZEUE3QlElxXsA9phyKSZNjTJDElGKENx46+4caGIW7/CnX9jp2Fth5I7uGce0nuCWJGlXacb6uwtLyulZcL21sbm3v2Lt7LRUlEpMmjlgkOwFShFBmpqRjqxJIgHjLSD0Xmtx+IVDQS93ocE4+jgaAhxUgbybcPeirhPoWVjrlO4E1Wq1V4CR3fLjs1Zwq4SNyclEGOhm9/9foRTjgRGjOkVNd1Yu2lSGqKGZmUeokiMcIjNCBdQwXiRHnpdIUJPDZKH4aRNEdoOFV/T6SIKzXmgenkSA/VvJeJ/3ndRIcXkpFnGgi8OyhMGFQRzDLA/apJFizsSEIS2r+CvEQSYS1Sa1kQnDnV14krdOa69Tcu7Ny/SqPowgOwRGoABecgzq4BQ3QBg8gmfwCt6sJ+vFerc+Zq0FK5/ZB39gf4A4liUfw=</latexit> <latexit sha1_base64="+hL14ZS6P1maznpjne569gjGrlQ=">ACAnicbVDLSgMxFM3UV62vUVfiJliEdmGZEUE3QlElxXsA9phyKSZNjTJDElGKENx46+4caGIW7/CnX9jp2Fth5I7uGce0nuCWJGlXacb6uwtLyulZcL21sbm3v2Lt7LRUlEpMmjlgkOwFShFBmpqRjqxJIgHjLSD0Xmtx+IVDQS93ocE4+jgaAhxUgbybcPeirhPoWVjrlO4E1Wq1V4CR3fLjs1Zwq4SNyclEGOhm9/9foRTjgRGjOkVNd1Yu2lSGqKGZmUeokiMcIjNCBdQwXiRHnpdIUJPDZKH4aRNEdoOFV/T6SIKzXmgenkSA/VvJeJ/3ndRIcXkpFnGgi8OyhMGFQRzDLA/apJFizsSEIS2r+CvEQSYS1Sa1kQnDnV14krdOa69Tcu7Ny/SqPowgOwRGoABecgzq4BQ3QBg8gmfwCt6sJ+vFerc+Zq0FK5/ZB39gf4A4liUfw=</latexit> <latexit sha1_base64="+hL14ZS6P1maznpjne569gjGrlQ=">ACAnicbVDLSgMxFM3UV62vUVfiJliEdmGZEUE3QlElxXsA9phyKSZNjTJDElGKENx46+4caGIW7/CnX9jp2Fth5I7uGce0nuCWJGlXacb6uwtLyulZcL21sbm3v2Lt7LRUlEpMmjlgkOwFShFBmpqRjqxJIgHjLSD0Xmtx+IVDQS93ocE4+jgaAhxUgbybcPeirhPoWVjrlO4E1Wq1V4CR3fLjs1Zwq4SNyclEGOhm9/9foRTjgRGjOkVNd1Yu2lSGqKGZmUeokiMcIjNCBdQwXiRHnpdIUJPDZKH4aRNEdoOFV/T6SIKzXmgenkSA/VvJeJ/3ndRIcXkpFnGgi8OyhMGFQRzDLA/apJFizsSEIS2r+CvEQSYS1Sa1kQnDnV14krdOa69Tcu7Ny/SqPowgOwRGoABecgzq4BQ3QBg8gmfwCt6sJ+vFerc+Zq0FK5/ZB39gf4A4liUfw=</latexit> Are the answers to my clicker questions random? X ( X i − E ( X i )) = 0 i 1 1 1 0.5 0 0 -0.5 -1 -1.5 -2 -2 a b c d 92

  79. <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> Are the answers to my clicker questions random? X ( X i − E ( X i )) 2 i 0.5 0.44 0.375 0.25 0.11 0.11 0.125 0 0 a b c d 93

  80. <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> <latexit sha1_base64="lfXAjwp0o8g6YPzeLGpW4AVYHYk=">ACAHicbVDLSgMxFM3UV62vURcu3ASL0C4sM0XQZVElxXsA9pxyKSZNjTJDElGKEM3/obF4q49TPc+Tdm2lo64HkHs65l+SeIGZUacf5tgorq2vrG8XN0tb2zu6evX/QVlEiMWnhiEWyGyBFGBWkpalmpBtLgnjASCcYX2d+5FIRSNxrycx8TgaChpSjLSRfPuorxLuU1jpmusM3mS1Wn2o+3bZqTkzwGXi5qQMcjR9+6s/iHDCidCYIaV6rhNrL0VSU8zItNRPFIkRHqMh6RkqECfKS2cLTOGpUQYwjKQ5QsOZ+nsiRVypCQ9MJ0d6pBa9TPzP6yU6vPRSKuJE4HnD4UJgzqCWRpwQCXBmk0MQVhS81eIR0girE1mJROCu7jyMmnXa65Tc+/Oy42rPI4iOAYnoAJcAEa4BY0QtgMAXP4BW8WU/Wi/VufcxbC1Y+cwj+wPr8AV8OlE4=</latexit> Are the answers to my clicker questions random? X ( X i − E ( X i )) 2 i 0.5 Thoughts? 0.44 0.375 0.25 0.11 0.11 0.125 0 0 a b c d 94

  81. <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> Are the answers to my clicker questions random? ( X i − E ( X i )) 2 X E ( X i ) i 0.16 0.148 0.12 0.08 0.037 0.037 0.04 0 0 a b c d 95

  82. <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> Are the answers to my clicker questions random? ( X i − E ( X i )) 2 X E ( X i ) i 96

  83. <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> Are the answers to my clicker questions random? ( X i − E ( X i )) 2 X E ( X i ) i 97

  84. <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> Are the answers to my clicker questions random? ( X i − E ( X i )) 2 X E ( X i ) i 98

  85. <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> Chi Squared Test ( X i − E ( X i )) 2 X E ( X i ) i Γ ( k/ 2) γ ( k 1 2 , x 2 ) 99

  86. <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="QD5Ua/6zG50qC5eqf+NtxgMtVBA=">ACIHicbVDLSsNAFJ3UV62vqEs3g0VoQWpShLosutBlBfuAJpTJdNIOmUnCzEQsIZ/ixl9x40IR3enXOG260NYLA+dxL3fu8WJGpbKsL6Owsrq2vlHcLG1t7+zumfsHRklApM2jlgkeh6ShNGQtBVjPRiQRD3GOl6wdXU794TIWkU3qlJTFyORiH1KUZKSwOz4fgC4dTOUucacY5gJTirVzNnlJPcDbK0np3CnDxMSXVglq2aNSu4DOw5KIN5tQbmpzOMcMJqDBDUvZtK1ZuioSimJGs5CSxAgHaET6GoaIE+mswMzeKVIfQjoV+o4Ez9PZEiLuWEe7qTIzWi95U/M/rJ8q/cFMaxokiIc4X+QmDKoLTtOCQCoIVm2iAsKD6rxCPkY5B6UxLOgR78eRl0KnXbKtm356Xm5fzOIrgCByDCrBAzTBDWiBNsDgETyDV/BmPBkvxrvxkbcWjPnMIfhTxvcPj86imQ=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> <latexit sha1_base64="+nxFYlAEC9qbXN+2+0COtMVmbg=">ACD3icbZBNS8MwGMdTX+d8q3r0EhzKdnC0Q9DjUASPE9wLrLWkWbqFJW1JUmGUfgMvfhUvHhTx6tWb38Z060E3Hwj58f8/D8nz92NGpbKsb2NpeWV1b20Ud7c2t7ZNf2OzJKBCZtHLFI9HwkCaMhaSuqGOnFgiDuM9L1x1e530gQtIovFOTmLgcDUMaUIyUljzxJEJ9yh0AoFwWu1pPIX+V2r3TeydIaZ1asujUtuAh2ARVQVMszv5xBhBNOQoUZkrJvW7FyUyQUxYxkZSeRJEZ4jIakrzFEnEg3ne6TwWOtDGAQCX1CBafq74kUcSkn3NedHKmRnPdy8T+vn6jgwk1pGCeKhHj2UJAwqCKYhwMHVBCs2EQDwoLqv0I8QjoZpSMs6xDs+ZUXodOo21bdvj2rNC+LOErgEByBKrDBOWiCG9ACbYDBI3gGr+DNeDJejHfjY9a6ZBQzB+BPGZ8/rjag=</latexit> Chi Squared Test ( X i − E ( X i )) 2 X E ( X i ) i { cdf that we can compute Γ ( k/ 2) γ ( k 1 2 , x explicitly 2 ) 100

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