Chapter 11 Hypothesis Testing II

• Chapter 11

• Hypothesis Testing II

Two Sample Tests

Matched Pairs

• Tests Means of 2 Related Populations

• Paired or matched samples
• Repeated measures (before/after)
• Use difference between paired values:

• Assumptions:

• Both Populations Are Normally Distributed

Test Statistic: Matched Pairs

• The test statistic for the mean difference is a t value, with

• n – 1 degrees of freedom:

Decision Rules: Matched Pairs

Assume you send your sales people to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data:

• Assume you send your sales people to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data:

Has the training made a difference in the number of complaints (at the  = 0.01 level)?

• Has the training made a difference in the number of complaints (at the  = 0.01 level)?

Difference Between Two Means

• Different data sources

• Unrelated
• Independent
• Sample selected from one population has no effect on the sample selected from the other population

Difference Between Two Means

σx2 and σy2 Known

σx2 and σy2 Known

Test Statistic, σx2 and σy2 Known

Hypothesis Tests for Two Population Means

Decision Rules

σx2 and σy2 Unknown, Assumed Equal

σx2 and σy2 Unknown, Assumed Equal

Test Statistic, σx2 and σy2 Unknown, Equal

σx2 and σy2 Unknown, Assumed Unequal

σx2 and σy2 Unknown, Assumed Unequal

Test Statistic, σx2 and σy2 Unknown, Unequal

Decision Rules

Pooled Variance t Test: Example

• You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data:

• NYSE NASDAQ Number 21 25

• Sample mean 3.27 2.53

• Sample std dev 1.30 1.16

Calculating the Test Statistic

Solution

• H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)

• H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)

•  = 0.05

• df = 21 + 25 - 2 = 44

• Critical Values: t = ± 2.0154

• Test Statistic:

Two Population Proportions

Two Population Proportions

• The random variable

• is approximately normally distributed

Test Statistic for Two Population Proportions

Decision Rules: Proportions

Example: Two Population Proportions

• Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?

• In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes

• Test at the 0.05 level of significance

The hypothesis test is:

• The hypothesis test is:

Example: Two Population Proportions

Hypothesis Tests of one Population Variance

• If the population is normally distributed,

Confidence Intervals for the Population Variance

Decision Rules: Variance

Hypothesis Tests for Two Variances

Hypothesis Tests for Two Variances

Test Statistic

Decision Rules: Two Variances

Example: F Test

• You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data:

• NYSE NASDAQ Number 21 25

• Mean 3.27 2.53

• Std dev 1.30 1.16

• Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.10 level?

F Test: Example Solution

• Form the hypothesis test:

• H0: σx2 = σy2 (there is no difference between variances)
• H1: σx2 ≠ σy2 (there is a difference between variances)

F Test: Example Solution

• The test statistic is:

Two-Sample Tests in EXCEL

• For paired samples (t test):

• Tools | data analysis… | t-test: paired two sample for means

• For independent samples:

• Independent sample Z test with variances known:

• Tools | data analysis | z-test: two sample for means

• For variances…

• F test for two variances:

• Tools | data analysis | F-test: two sample for variances

Two-Sample Tests in PHStat

Sample PHStat Output

Sample PHStat Output