Independent Samples: Comparing Proportions Lecture 38 Section 11.5 Wed, Nov 7, 2007

The Sampling Distribution of p1^ – p2^ p1^ is N(p1, 1), where p2^ is N(p2, 2), where

Statistical Fact #1 For any two random variables X and Y,

Statistical Fact #2 Furthermore, if X and Y are both normal, then X – Y is normal. That is, if X is N(X, X) and Y is N(Y, Y), then

The Sampling Distribution of p1^ – p2^ Therefore, where

The Test Statistic Therefore, the test statistic would be if we knew the values of p1 and p2. We will approximate them with p1^ and p2^.

The Test Statistic Therefore, the test statistic would be except that…

Pooled Estimate of p The null hypothesis is typically H0: p1 = p2 Under that assumption, p1^ and p2^ are both estimators of a common value, which we will call p.

Pooled Estimate of p Rather than use either p1^ or p2^ alone to estimate p, we will use a “pooled” estimate. The pooled estimate is the proportion that we would get if we pooled the two samples together.

Pooled Estimate of p The “Batting-Average” Formula:

Case Study 13 In the survey, we had 240 males (48%) and 260 females (52%). 41% of the males, or 98 males, said Wilder is doing good or excellent. 37% of the females, or 96 females, said he is doing good or excellent. Altogether, 194 people out of 500, or 38.8%, said he is doing good or excellent.

The Standard Deviation of p1^ – p2^ This leads to a better estimator of the standard deviation of p1^ – p2^.

Case Study 13 Compute

Caution If the null hypothesis does not say H0: p1 = p2 then we should not use the pooled estimate p^, but should use the unpooled estimate

The Test Statistic So the test statistic is where

The Value of the Test Statistic Compute z:

The p-value, etc. Compute the p-value: P(Z > 0.9170) =0.1796. Reject H0. Equal proportions of men and women believe that Mayor Wilder is doing a good or excellent job.

Case Study 13 Continued Do equal proportions of whites and blacks believe that Mayor Wilder is doing a good or excellent job? Do equal proportions of Republicans and Democrats believe that Mayor Wilder is doing a good or excellent job? City Hall turmoil: RT-D poll.

TI-83 – Testing Hypotheses Concerning p1^ – p2^ Press STAT > TESTS > 2-PropZTest... Enter x1, n1 x2, n2 Choose the correct alternative hypothesis. Select Calculate and press ENTER.

TI-83 – Testing Hypotheses Concerning p1^ – p2^ In the window the following appear. The title. The alternative hypothesis. The value of the test statistic z. The p-value. p1^. p2^. The pooled estimate p^. n1. n2.

Example Work Case Study 13 again, using the TI-83.

Confidence Intervals for p1^ – p2^ The formula for a confidence interval for p1^ – p2^ is Caution: Note that we do not use the pooled estimate for p^.

TI-83 – Confidence Intervals for p1^ – p2^ Press STAT > TESTS > 2-PropZInt… Enter x1, n1 x2, n2 The confidence level. Select Calculate and press ENTER.

TI-83 – Confidence Intervals for p1^ – p2^ In the window the following appear. The title. The confidence interval. p1^. p2^. n1. n2.

Example Find a 95% confidence interval for the difference between the proportions of whites and blacks who believe that Mayor Wilder is doing a good or excellent job.