The Practice of StatisticsThird Edition

Chapter 13:Comparing Two Population Parameters

Copyright © 2008 by W. H. Freeman & Company

Daniel S. Yates

Section 13.2Comparing 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 phat1- phat2

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) ParameterIdentify the populations and the parameters.

2) Check the conditions for BOTH proportions.3) Calculations4) 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.

1 2c

1 2

X Xcount of successes in both samples combinedp̂

count of individuals in both samples combined n n

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) HypothesesIdentify the parameters. State the

hypotheses and the procedure name.

Ho: p1= p2 or, equivalently Ho: p1- p2 = 0

Ha: p1 > p2 Ha: p1- p2 > 0

2) Check the conditions for BOTH proportions.3) Calculations4) 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 PressureDoes 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

7 -4 18 17 -3 -5 1 10 11 -2

Placebo Group

-1 12 -1 -3 3 -5 5 2 -11 -1 -3

Calcium and Blood Pressure

Summary Stats

Group Treatment n xbar s

1 Calcium 10 5.000 8.743

2 Placebo 11 -0.273 5.901

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) Calculations4) 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.

CalculationsUse 2-Samp T Int on calculatorFill in formula for CIEstimate ± (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.

2 21 2

1 2

1 2

s s(x x ) t *

n n

Interpretation

• We are 90% confident that the true mean difference in systolic blood pressure between the two groups is between -.4767 and 11.022.

• 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) Calculations4) Interpretation

1) HypothesesIdentify the parameters. State the

hypotheses and the procedure name.

Ho: μ1= μ 2 or, equivalently Ho: μ 1- μ 2 = 0

Ha: μ 1 > μ 2 Ha: μ 1- μ 2 > 0

2) Check the conditions for BOTH samples.3) Calculations4) Interpretation