Ch 10 Comparing Two Proportions Target Goal: I can determine the significance of a two sample proportion. 10.1b h.w: pg 623: 15, 17, 21, 23

 If we want to compare two populations or compare the responses to two treatments from independent samples which involve the mean of a quantitative variable, we look at a two- sample method.

Comparing Proportions of Successes in Two Groups  The null hypothesis is that there is no difference between the two parameters.

The Alternative Hypothesis could be that (two-sided)(one-sided)(one-sided)  The alternative says what kind of difference we expect.

Pooled Sample Proportion  When the observations come from a single population; instead of estimating p 1 and p 2 separately, we pool the two samples (combined) and use the overall sample proportion the estimate the single parameter p. = Count of successes in both samples combined Count of observations in both samples combined Count of observations in both samples combined

Check your assumptions about a proportion! 1) Random: Both samples formed from a random sample or by two groups from a randomized experiment. 2) Independently chosen samples. (A randomized sample taken covers independence) Population  10n; population is at least 10 times as large as the sample. 3. Normal:  10 for a 2 sample hypothesis test  10 for a 2 sample hypothesis test  10 for 2 sample a C.I.  10 for 2 sample a C.I.

If these assumptions hold, then the difference in sample proportions is an unbiased estimator of the difference in population proportions,  The mean of is equal to.

The variance of is the sum of the variances of  Recall that the variances add but the standard deviations do not.

When sample sizes are large, the distribution of is approximately normal. Many comparative studies start with just one sample and then divide into two groups based on the data gathered from the subjects. Use the two sample z procedures.

 Replace the population proportions with the sample proportions to find SE.

Significance Tests for Comparing Two Proportions  Significance tests help us to decide if the effect we see in the samples is really there in the proportions.  The null hypothesis is that there is no difference between the two parameters.

 In order to standardize, subtract the mean and divide by the standard error (SE):  Use this test statistic to carry out a test of significance.  Note: most 2-sample problems are pooled. pooled

Ex: Cholesterol and Heart Attacks  High levels of cholesterol in the blood are associated with higher risk of heart attacks. Study: Middle age men were assigned at random to one of two treatments: 2051 men took the drug gemfibrozil to reduce their cholesterol levels, and a control group of 2030 men took a placebo.  During the next five years, 56 men in the gemfibrozil group and 84 men in the placebo group had heart attacks.

The sample proportion that had heart attacks are: (gemfibrozil group) (placebo group)  Is the apparent benefit of gemfibrozil statistically significant?

Step 1.State Identify the population of interest and the parameter you want to draw a conclusion about.  We hope to show that gemfibrozil reduces heart attacks by comparing two proportions of middle age men. : There is no difference in the reduction of heart attacks for middle age men taking gemfibrozil compared to the placebo.

 There is a reduction in the number of heart attacks of middle age men taking gemfibrozil compared to the middle age men taking the placebo.

Step 2. Plan If the conditions are met, we should perform a two- sample z test for.  Since both samples come from a single population, we will pool the samples.  The pooled proportion of heart attacks for the two groups in the Helinski Heart study is: = Count of heart attacks in both samples combined = Count of heart attacks in both samples combined Count of subjects in both samples combined Count of subjects in both samples combined

Verify the conditions.  Random: subjects randomly assigned  Independent: Yes, due to random assignment, these groups can be viewed as independent. Individual observations are also independent. Knowing whether one subject has a heart attack gives no information on another subject. Individual observations are also independent. Knowing whether one subject has a heart attack gives no information on another subject.

Verify the conditions.  Normal: ;  10. ;  10. pooled

Step 3. Carry out the inference procedure  The z statistic is

Determine the p-value  normalcdf(-E99,-2.47) = Step 4: Conclude - Interpret your results in the context of the problem. Since P < 0.01, the results are significant at the α= 0.01 level. We reject H o and conclude that gemfibrozil reduced the rate of heart attacks in middle age men.

Verify by calculator:  On the TI-83+, use 2-PropZInt and 2- PropZTest to construct confidence intervals and perform significance tests.  Stat:tests:2-propZTest:  In class FR: 2009B #3  Read pg Gives you z score using pooled proportions.