Hypothesis Tests for Population Proportions Bernd Schr oder logo1 - - PowerPoint PPT Presentation

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Hypothesis Tests for Population Proportions

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test)

✲ µ0 α

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test)

✲ µ0 α Upper tail test for µ≤µ0:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test)

✲ µ0 α Upper tail test for µ≤µ0: Tail probability is ≤α (small) if µ≤µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test)

✲ µ0 α

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test)

✲ µ0 α Lower tail test for µ≥µ0:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test)

✲ µ0 α Lower tail test for µ≥µ0: Tail probability is ≤α (small) if µ≥µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test) Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲ µ0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲ µ0

α 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲ µ0

α 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲ µ0

α 2 α 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲ µ0

α 2 α 2

Two tailed test for µ = µ0:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To test H0 : p = p0 ...

... as long as np0 ≥ 10 and n(1−p0) ≥ 10. (So that the normal approximation for the binomial distribution works.)

• 1. Test statistic: z =

ˆ p−p0

• p0(1−p0)/n
• 2. Alternative hypotheses and rejection regions.

◮ For Ha : p > p0 use z ≥ zα (upper tailed test) ◮ For Ha : p < p0 use z ≤ −zα (lower tailed test) ◮ For Ha : p = p0 use |z| ≥ z α 2 (two-tailed test)

✲ µ0

α 2 α 2

Two tailed test for µ = µ0: Tail probability is α (small) if µ=µ0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region: z > z0.01 ≈ 2.33.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region: z > z0.01 ≈ 2.33. Substitute values into test statistic:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region: z > z0.01 ≈ 2.33. Substitute values into test statistic: z = ˆ p−p0

• p0(1−p0)/n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region: z > z0.01 ≈ 2.33. Substitute values into test statistic: z = ˆ p−p0

• p0(1−p0)/n

=

50 500 −0.05

• 0.05(1−0.05)/500

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region: z > z0.01 ≈ 2.33. Substitute values into test statistic: z = ˆ p−p0

• p0(1−p0)/n

=

50 500 −0.05

• 0.05(1−0.05)/500

≈ 5.13

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region: z > z0.01 ≈ 2.33. Substitute values into test statistic: z = ˆ p−p0

• p0(1−p0)/n

=

50 500 −0.05

• 0.05(1−0.05)/500

≈ 5.13 Decide if to reject or not.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that, on average, 5% of its flights

are delayed each day. On a given day, of 500 flights, 50 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. Parameter of interest: p0, average proportion of delayed flights. Null hypothesis: H0 : p0 = 5% = 0.05 Alternative hypothesis: Ha : p0 > 0.05 (the main concern is a larger average proportion than what is claimed). Test statistic: z = ˆ p−p0

• p0(1−p0)/n

Rejection region: z > z0.01 ≈ 2.33. Substitute values into test statistic: z = ˆ p−p0

• p0(1−p0)/n

=

50 500 −0.05

• 0.05(1−0.05)/500

≈ 5.13 Decide if to reject or not. Reject.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error) = P(X ≥ c|X ∼ Bin(n,p0))

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error) = P(X ≥ c|X ∼ Bin(n,p0)) = 1−P(X < c|X ∼ Bin(n,p0))

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error) = P(X ≥ c|X ∼ Bin(n,p0)) = 1−P(X < c|X ∼ Bin(n,p0)) = 1−B(c−1;n;p0)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error) = P(X ≥ c|X ∼ Bin(n,p0)) = 1−P(X < c|X ∼ Bin(n,p0)) = 1−B(c−1;n;p0)

• 4. Type II errors.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error) = P(X ≥ c|X ∼ Bin(n,p0)) = 1−P(X < c|X ∼ Bin(n,p0)) = 1−B(c−1;n;p0)

• 4. Type II errors.

β(p′)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error) = P(X ≥ c|X ∼ Bin(n,p0)) = 1−P(X < c|X ∼ Bin(n,p0)) = 1−B(c−1;n;p0)

• 4. Type II errors.

β(p′) = P(X < c|X ∼ Bin(n,p′))

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

To Test H0 : p = p0 Versus Ha : p > p0 ...

... for small sample size, use the binomial distribution directly.

• 1. Test statistic: Number x of favorable outcomes.
• 2. Rejection region: x ≥ c, where c is an integer.
• 3. Type I errors.

P(type I error) = P(X ≥ c|X ∼ Bin(n,p0)) = 1−P(X < c|X ∼ Bin(n,p0)) = 1−B(c−1;n;p0)

• 4. Type II errors.

β(p′) = P(X < c|X ∼ Bin(n,p′)) = B(c−1;n;p′)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are delayed.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. α = 1−B(c−1;n;p0)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. α = 1−B(c−1;n;p0) 0.01 = 1−B(c−1;25;0.05)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. α = 1−B(c−1;n;p0) 0.01 = 1−B(c−1;25;0.05) B(c−1;25;0.05) = 0.99

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. α = 1−B(c−1;n;p0) 0.01 = 1−B(c−1;25;0.05) B(c−1;25;0.05) = 0.99 c−1 = 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. α = 1−B(c−1;n;p0) 0.01 = 1−B(c−1;25;0.05) B(c−1;25;0.05) = 0.99 c−1 = 3 c = 4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Test the hypothesis that the average proportion of

delayed flights is 5% at the 0.01 level. α = 1−B(c−1;n;p0) 0.01 = 1−B(c−1;25;0.05) B(c−1;25;0.05) = 0.99 c−1 = 3 c = 4 Conclusion: Do not reject.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are delayed.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Compute the probability of a type II error when testing

the hypothesis that the average proportion of delayed flights is 5% at the 0.01 level and the actual proportion is 10%.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Compute the probability of a type II error when testing

the hypothesis that the average proportion of delayed flights is 5% at the 0.01 level and the actual proportion is 10%. β = B(c−1;n;p′)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Compute the probability of a type II error when testing

the hypothesis that the average proportion of delayed flights is 5% at the 0.01 level and the actual proportion is 10%. β = B(c−1;n;p′) = B(3;25;0.1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions

logo1 Large Sample Size Small Sample Size

• Example. An airline claims that on average 5% of its flights

are delayed each day. On a given day, of 25 flights, 3 are

• delayed. Compute the probability of a type II error when testing

the hypothesis that the average proportion of delayed flights is 5% at the 0.01 level and the actual proportion is 10%. β = B(c−1;n;p′) = B(3;25;0.1) = 0.764

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Hypothesis Tests for Population Proportions