The Practice of Statistics Third Edition Chapter 13: Comparing Two Population Parameters Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

Section 13.2 Comparing Two Proportions By using inference procedures on two proportions from the two groups of interest, we can compare two populations. Remember that inference procedures for a single proportion relied on the sampling distribution of phat. Similarly, inference procedures for two proportions rely on the sampling distribution of phat1- phat2.

The rules of Random Variables are used to get the center and spread of the Sampling Distribution of phat 1 - phat 2

Example 13.9 How much does Preschool Help? 49 out of 61 who did not attend preschool needed social services as adults 38 out of 62 who attended preschool needed social services as adults Construct and interpret a 95% Confidence Interval that for the difference between the proportions (did not attend – attended).

Steps 1)Parameter Identify the populations and the parameters. 2)Check the conditions for BOTH proportions. 3) Calculations 4) Interpretation

Interpretation We are 95% confident that the percent needing social services is somewhere between 3.3% and 34.7% lower among people who attended preschool. OR We are 95% confident that the percent needing social services is somewhere between 3.3% and 34.7% higher among people who did not attend preschool.

Combined Sample Proportion This is the overall sample proportion used to estimate the single population parameter p.

How much does Preschool Help? Again! 49 out of 61 who did not attend preschool needed social services as adults 38 out of 62 who attended preschool needed social services as adults Is there strong evidence to support the claim that those that did not attend preschool have greater need for social services as adults?

1)Hypotheses Identify the parameters. State the hypotheses and the procedure name. H o : p 1 = p 2 or, equivalently H o : p 1 - p 2 = 0 H a : p 1 > p 2 H a : p 1 - p 2 > 0 2) Check the conditions for BOTH proportions. 3) Calculations 4) Interpretation

Interpretation The P-value,.0102, tell us that it is very unlikely that we would obtain a difference in sample proportions as large as we did if the null hypothesis is true. Therefore, we reject the null hypothesis and conclude that there is strong evidence to support the claim that those that did not attend preschool have greater need for social services as adults

Computer Output for 2 Proportion Z-Test

Note that this is not the same as matched pairs. There is no matching here. The samples are independent and may even be different sizes. Section 13.1 Comparing Two Means

Matched Pairs vs. Two Sample P.781 Ex 13.1

Calcium and Blood Pressure Does increasing the amount of calcium in our diet reduce blood pressure? Randomized Comparative Experiment Results The response variable is the decrease in systolic (top number) bp. Calcium Group Placebo Group

Calcium and Blood Pressure Summary Stats GroupTreatmentnxbars 1Calcium Placebo The calcium group shows a 5 point decrease in bp. The placebo group shows a slight increase in bp.

Calcium and Blood Pressure We want to calculate a 90% CI for the differences in systolic blood pressure between the two groups. 1) Parameter: Identify the populations and the parameters. 2) Check the conditions for BOTH samples. 3) Calculations 4) Interpretation

SRS Since the 21 subjects are not an SRS, it may be difficult to generalize our findings to all healthy black men. However, the random assignment of subjects to treatments should help ensure that any significant differences in mean bp between the two groups is due to the treatments.

Checking for Non-Normality with NPP’s There are no outliers and no departures from Normality that prevent use of t procedures.

Independence Because of the randomization, we are willing to regard the calcium and placebo groups as two independent samples. We are not sampling without replacement from a population of interest in this case.

Calculations Use 2-Samp T Int on calculator Fill in formula for CI Estimate ± (Critical Value)(St Dev of Statistic) Note that the calculator tells you the df and it is not a whole number! You do not have to figure out t*. Just leave it as t*. However, do note the df on your paper.

Interpretation We are 90% confident that the true mean difference in systolic blood pressure between the two groups is between and Would we reject the null hypothesis that the two groups have the same mean systolic blood pressures?

Calcium and Blood Pressure Now, we want to test if the differences in systolic blood pressure between the two groups is significant. 1) Parameter: Identify the populations and the parameters. 2) Check the conditions for BOTH samples. 3) Calculations 4) Interpretation

1)Hypotheses Identify the parameters. State the hypotheses and the procedure name. H o : μ 1 = μ 2 or, equivalently H o : μ 1 - μ 2 = 0 H a : μ 1 > μ 2 H a : μ 1 - μ 2 > 0 2) Check the conditions for BOTH samples. 3) Calculations 4) Interpretation