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Chapter 7.2 & 7.3 Z-scores & Probabilities Learning Objectives At the end of this lecture, the student should be able to: Explain how to convert an x to a z-score. Show how to look up a z-score in a z table. Explain how

  1. Chapter 7.2 & 7.3 Z-scores & Probabilities

  2. Learning Objectives At the end of this lecture, the student should be able to: • Explain how to convert an x to a z-score. • Show how to look up a z-score in a z table. • Explain how to find the probability of an x falling between two values on a normal distribution. • Describe how to use the z table to look up a z corresponding to a percentage. • Describe how to use the formula to calculate x from a z-score

  3. Introduction • Z-score and the standard normal distribution • Z-score probabilities • Using the z table to answer harder questions • Calculating x from z • Using z-score and probabilities correctly Photograph by Dirk Beyer

  4. What is a Z-Score? Introduction to Standard Normal Distribution

  5. Remember the Empirical Rule? • Required normal distribution • Worked well for the cutpoints available • What about in-between?

  6. Remember the Empirical Rule? • Required normal distribution • Worked well for the cutpoints available • What about in-between? • Notice the numbers along the bottom • -3, -2, -1, then µ (which has no number, or 0), then 1, 2, 3

  7. Z is the Standard Normal Distribution

  8. Z-scores • Every value on a normal distribution (every “x”) can be converted to a z-score. • You must know the following to use formula: • The “x” – what you want to convert to z • The µ of the distribution • The σ of the distribution

  9. Z-scores • Every value on a normal z = x - μ x = zϬ + μ distribution (every “x”) can Ϭ be converted to a z-score. • You must know the following to use formula: • The “x” – what you want to convert to z • The µ of the distribution • The σ of the distribution

  10. Z-scores: Smart Friend Example µ = 65.5 σ = 14.5 • Remember our n=100 z = x - μ x = zϬ + μ students? Ϭ • Let’s say your friend got a 90. What is the z-score for 90?

  11. Z-scores: Smart Friend Example µ = 65.5 σ = 14.5 z = x - μ x = zϬ + μ Ϭ 22 36.5 51 65.5 80 94.5 109 90?

  12. Z-scores: Smart Friend Example µ = 65.5 σ = 14.5 • Remember our n=100 z = x - μ x = zϬ + μ students? Ϭ • Let’s say your friend got a 90. What is the z-score for 90? • x=90 • µ = 65.5 • σ = 14.5 z=0 x=65.5

  13. Z-scores: Smart Friend Example µ = 65.5 σ = 14.5 • Remember our n=100 z = x - μ x = zϬ + μ students? Ϭ • Let’s say your friend got a 90. What is the z-score for 90? • x=90 • µ = 65.5 • σ = 14.5 z=0 z=1.69 • x=65.5 (90-65.5)/14.5 = 1.69 x=90

  14. Z-scores: Not-so-smart Friend Example µ = 65.5 σ = 14.5 • n=100 z = x - μ x = zϬ + μ Ϭ • Let’s say your other friend got a 50. What is the z- score for 50? z=0 x=65.5

  15. Z-scores: Not-so-smart Friend Example µ = 65.5 σ = 14.5 z = x - μ x = zϬ + μ Ϭ 22 36.5 51 65.5 80 94.5 109 50?

  16. Z-scores: Not-so-smart Friend Example µ = 65.5 σ = 14.5 • n=100 z = x - μ x = zϬ + μ Ϭ • Let’s say your other friend got a 50. What is the z- score for 50? • x=50 • µ = 65.5 • σ = 14.5 • (50-65.5)/14.5 = -1.07 z=0 x=65.5 • It is negative because it is below µ

  17. Z-scores: Not-so-smart Friend Example µ = 65.5 σ = 14.5 • n=100 z = x - μ x = zϬ + μ Ϭ • Let’s say your other friend got a 50. What is the z- score for 50? • x=50 • µ = 65.5 • σ = 14.5 • (50-65.5)/14.5 = -1.07 z=0 z=-1.07 x=65.5 • It is negative because it is x=50 below µ

  18. Z-score Probabilities Using the Z table

  19. Remember “Probability” from the Empirical Rule? Question: What is the probability I will select a student with a score between 36.5 and 51? Answer: 13.5% But what if you have z-scores of 1.69 (smart friend) and 22 36.5 51 65.5 80 94.5 109 -1.07 (not-so-smart friend)?

  20. Questions about Z-Score Probabilities µ = 65.5 σ = 14.5 • What is the probability that z = x - μ x = zϬ + μ students scored above the smart Ϭ friend? • In other words – what is the area under the curve from z=1.69 all the way up? z=0 z=1.69 x=65.5 x=90

  21. Questions about Z-Score Probabilities µ = 65.5 σ = 14.5 • What is the probability that z = x - μ x = zϬ + μ students scored above the smart Ϭ friend? • In other words – what is the area under the curve from z=1.69 all the way up? • What is the probability that students scored below the not-so- smart friend? • In other words – what is the area under the curve from z=- z=0 z=1.69 1.07 all the way down? z=-1.07 x=65.5 x=90 x=50

  22. Questions about Z-Score Probabilities µ = 65.5 σ = 14.5 • What is the probability that z = x - μ x = zϬ + μ students scored above the smart Ϭ friend? • In other words – what is the area under the curve from z=1.69 all the way up? • What is the probability that students scored below the not-so- smart friend? • In other words – what is the area under the curve from z=- z=0 z=1.69 1.07 all the way down? z=-1.07 x=65.5 x=90 x=50 We will look these up using the Z table.

  23. How to Use the Z Table µ = 65.5 σ = 14.5 • First, figure out what area z = x - μ x = zϬ + μ Ϭ you want. • What is the probability that students scored below the not-so-smart friend (z=-1.07, x=50)? • For areas to the left of a specified z value, use the z=0 table entry directly . z=-1.07 x=65.5 x=50

  24. How to Use the Z Table µ = 65.5 σ = 14.5 • First, figure out what area z = x - μ x = zϬ + μ Ϭ you want. • What is the probability that students scored below the not-so-smart friend (z=-1.07, x=50)? • For areas to the left of a specified z value, use the z=0 table entry directly . z=-1.07 x=65.5 x=50 p = 0.1423. The probability is 14.23%.

  25. How to Use the Z Table µ = 65.5 σ = 14.5 • Let’s do the smart friend’s z = x - μ x = zϬ + μ probability. Ϭ • What is the probability that students scored above the smart friend (x=90, z=1.69)? • For areas to the right of a specified z value, either: • Look up in table, then subtract result from 1, or z=0 z=1.69 • Use the opposite z (-1.69) x=65.5 x=90

  26. How to Use the Z Table µ = 65.5 σ = 14.5 • Let’s do the smart friend’s z = x - μ x = zϬ + μ probability. Ϭ • What is the probability that students scored above the smart friend (x=90, z=1.69)? • For areas to the right of a specified z value, either: • Look up in table, then subtract result from 1, or z=0 z=1.69 • Use the opposite z (-1.69) x=65.5 x=90

  27. How to Use the Z Table µ = 65.5 σ = 14.5 • Let’s do the smart friend’s z = x - μ x = zϬ + μ probability. Ϭ • What is the probability that students scored above the smart friend (x=90, z=1.69)? • For areas to the right of a specified z value, either: • Look up in table, then subtract result from 1, or z=0 z=1.69 • Use the opposite z (-1.69) x=65.5 x=90 p = 0.9545. 1 – 0.9545 = 0.0455, or 4.55%

  28. How to Use the Z Table µ = 65.5 σ = 14.5 • Let’s do the smart friend’s z = x - μ x = zϬ + μ probability. Ϭ • What is the probability that students scored above the smart friend (x=90, z=1.69)? • For areas to the right of a specified z value, either: • Look up in table, then subtract result from 1, or z=0 z=1.69 • Use the opposite z (-1.69) x=65.5 x=90

  29. How to Use the Z Table µ = 65.5 σ = 14.5 • Let’s do the smart friend’s z = x - μ x = zϬ + μ probability. Ϭ • What is the probability that students scored above the smart friend (x=90, z=1.69)? • For areas to the right of a specified z value, either: • Look up in table, then subtract result from 1, or z=0 z=1.69 • Use the opposite z (-1.69) x=65.5 x=90 p = 0.0455, or 4.55%

  30. Harder Questions More on Probabilities and Z Table

  31. Harder Questions µ = 65.5 σ = 14.5 • What if you are looking at a z = x - μ x = zϬ + μ probability between two scores Ϭ – such as the probability the students will score between 50 and 90? 22 36.5 51 65.5 80 94.5 109

  32. Calculating Probability Between Scores µ = 65.5 σ = 14.5 1. Note that you have x1 and x2 z = x - μ x = zϬ + μ (two x’s) Ϭ x1=50 x2=90

  33. Calculating Probability Between Scores µ = 65.5 σ = 14.5 1. Note that you have x1 and x2 z = x - μ x = zϬ + μ (two x’s) Ϭ 2. Calculate z1 and z2 z2=1.69 z1=-1.07 x2=90 x1=50

  34. Calculating Probability Between Scores µ = 65.5 σ = 14.5 1. Note that you have x1 and x2 z = x - μ x = zϬ + μ (two x’s) Ϭ 2. Calculate z1 and z2 3. For z1, find the probability to the left (below) z • Remember, direct probability from z table z2=1.69 z1=-1.07 x2=90 x1=50

  35. Calculating Probability Between Scores µ = 65.5 σ = 14.5 1. Note that you have x1 and x2 z = x - μ x = zϬ + μ (two x’s) Ϭ 2. Calculate z1 and z2 3. For z1, find the probability to the left (below) z • Remember, direct probability from z table 4. For z2, find the probability to the right (above) z • Use one of the 2 methods z2=1.69 z1=-1.07 x2=90 x1=50

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