Chapter 22: Comparing Two Proportions

Yet Another Standard Deviation (YASD) Standard deviation of the sampling distribution The variance of the sum or difference of two independent random variables is the sum of their variances

Standard Deviation of the Difference Between Two Proportions Proportions observed in independent random samples are independent Sample proportions:

Assumptions & Conditions Independence Assumption: Within each group, the data should be based on results for independent individuals Randomization: The data in each group should be drawn independently and at random from a homogeneous population or generated by a randomized comparative study

Assumptions & Conditions 10% condition: When the data are sampled without replacement, the sample should not exceed 10% of the population. Independent Samples condition: The two groups that are being compared must be independent of one another. Success/failure: Both samples are big enough that at least 10 successes and and least 10 failures have been observed.

The Sampling Distribution The sampling distribution model for a difference between two independent proportions: Provided that the sampled values are independent, the samples are independent, and the sample sizes are large enough

A Two-proportion z-interval When the conditions are met, find the confidence interval: The critical value depends on the confidence level, C, that you specify.

Intelligence – An Example Identify the parameter and choose a confidence level

Intelligence – An Example Check the conditions. Randomization: Gallup drew a random sample of U.S. adults 10%: the sample size for each groups was certainly less than the U.S. population of men and women Independent Samples: the sample of women and the sample of men are independent of each other

Intelligence – An Example Check the conditions. Success/Failure: Both samples exceed the minimum size.

Intelligence – An Example State the sampling distribution model: Under these conditions, the sampling distribution of the difference between the sample proportions is approximately Normal with a mean of p M – p F, the true difference between the population proportions. Find a two-proportion z-interval.

Intelligence – An Example Calculator

Intelligence – An Example Interpret the results: We are 95% confident that the proportion of American men that the attribute of “intelligent” applies more to men than to women is between 9% and 19% more than American women who think that,

Pooling Pooling combines the counts to get an overall proportion When we have counts for each group: When we have only proportions: Round to whole numbers

Two-proportion z-test The conditions for the two-proportion z-test are the same as for the two-proportion z- interval Test the hypothesis Because we hypothesize that the proportions are equal, pool to find

Two-proportion z-test Standard Error: Find the test statistic:

Snoring Rates – An Example Hypothesis: H O :There is no difference in snoring rates between those who are 18 – 29 years old and those who are 30 years old. H A : The rates are different.

Snoring Rates – An Example Check the conditions: Randomization: the patients were randomly selected and stratified by sex and region 10%: the number of adults surveyed is certainly less than 10% of the population. Independent samples: the two groups are independent of each other Success/failure: Younger group: 48 snored, 136 didn’t Older group: 318 snored, 493 didn’t

Snoring Rates – An Example State the null model and choose your method: Because the conditions are met, we can model the sampling distribution of the difference in proportions with a Normal model Perform a two-proportion z-test

Snoring Rates – An Example Calculator:

Snoring Rates – An Example State your conclusion: The P-value of says that if there really were no difference in snoring rates between the two age groups, then the difference observed would only happen 8 out of 10,000 times. This is rare enough for us to reject the null hypothesis of no difference and conclude that there is a difference between older and younger adults. It appears that older adults are more likely to snore.

What Can Go Wrong??? Don’t use two-sample proportion methods when the samples aren’t independent Make sure there is no relationship between the two groups When the assumption of independence is violated, this method gives wrong answers Don’t apply inference methods when you don’t have random samples Don’t interpret a significant difference in proportions causally.