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Chapter 10, sections 1 and 4 Two-sample Hypothesis Testing Test hypotheses for the difference between two independent population means ( standard deviations known or unknown) Use the F table to find critical F values Complete an F test for the difference between two variances

Difference Between Two Means Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ 1 – μ 2. Assumptions: Different data sources-- populations are Unrelated and Independent two samples are randomly and independently drawn from these populations. Sample selected from one population has no effect on the sample selected from the other population population distributions are normal or both sample sizes are  30

Possible Hypotheses Are: Upper-tail test: H 0 : μ 1 ≤ μ 2 H 1 : μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Lower-tail test: H 0 : μ 1  μ 2 H 1 : μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2  0 H 1 : μ 1 – μ 2 < 0 Two-tail test: H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0

Population standard deviations are known, σ 1 and σ 2 known, use Z test. The test statistic for μ 1 – μ 2, based on sample sample means, is: The Confidence interval for μ 1 – μ 2 is:

σ 1 and σ 2 Unknown Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed or both sample sizes are at least 30  Population variances are unknown but assumed equal, σ 1 and σ 2 unknown, but known to be equal  The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ  The test Statistic for μ 1 – μ 2 has a t distribution with a degree of freedom of (n 1+ n 2 -2):

Where: The confidence interval for μ 1 – μ 2 is: Example: Last week you were given a sample of 69 beer, 54 are U.S.-made and 15 are foreign-made. Additional information were provided with respect to price, calories, and percent alcohol content. Let’s assume the two populations are unrelated, independent, and approximately normally distributed with equal variance Assume that the two samples are independently drawn.

Samples Information:

Questions: 1.Is there evidence of a difference in mean calories of us and non-U.S. beers? 2.What is the 95% confidence interval for the difference in mean calories? 3.Are conclusions in 1 and 2 consistent? 4.U.S. beers have about 10% more alcohol than non-U.S. beers. 5.Is the assumption of equal population variances, that you used for 1 and 4 a valid assumption? 6.Is there evidence that there is less variation in price of imported beers than price of domestic beers

Hypothesis Tests for Variances Test of two population variances Hypotheses: H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 Two-tail test Lower-tail test Upper-tail test H 0 : σ 1 2  σ 2 2 H 1 : σ 1 2 < σ 2 2 H 0 : σ 1 2 ≤ σ 2 2 H 1 : σ 1 2 > σ 2 2 H 0 : σ 1 2 / σ 2 2 =1 H 1 : σ 1 2 / σ 2 2 ≠1 H 0 : σ 1 2 / σ 2 2  1 H 1 : σ 1 2 / σ 2 2 <1 H 0 : σ 1 2 / σ 2 2 ≤1 H 1 : σ 1 2 / σ 2 2 >1

The test statistic from samples is F-Distribution can take values from 0 to infinity It is a right-skewed distribution = Variance of Sample 1 n 1 - 1 = numerator degrees of freedom n 2 - 1 = denominator degrees of freedom = Variance of Sample 2

F0  /2 Do not reject H 0 FUFU FLFL  /2 1.Finding the critical lower and upper tail values 2.Find F U from the F table for n 1 – 1 numerator and n 2 – 1 denominator degrees of freedom 3.Find F L using the formula: Where F U* is from the F table with n 2 – 1 numerator and n 1 – 1 denominator degrees of freedom (i.e., switch the d.f. from F U ) H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2

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