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- Home | Text Table of Contents | Assignments | Calculator | Tools | Review | Glossary | Bibliography | System Requirements | Author's Homepage Chapter 21 Testing Equality of Two Percentages Chapter 19, "Hypothesis Testing: Does Chance Explain the Results?" introduced a conceptual framework for statistical hypothesis testing. Chapter 20, "Does Treatment Have an Effect?," presented important statistical considerations for determining whether a treatment has an effect. Treatment is meant loosely--it could be a drug, an advertising campaign, a car wax, a test preparation course, a fertilizer, etc . The best way to determine whether a treatment has an effect is to use the method of comparison in an experiment in which subjects are assigned at random to the treatment group or the control group . When the measurement of each subject can be represented by 0 or 1 ( e.g. , subject's condition improves or not, subject buys something or not, subject clicks a link or not, subject passes an exam or not), deciding whether the treatment has an effect is essentially testing the null hypothesis that two percentages are equal--which is the problem this chapter addresses. Different ways of drawing samples lead to different tests. In one sampling design (the randomization model ), the entire collection of subjects is allocated randomly between treatment and control, which makes the samples dependent . Conditioning on the total number of ones in the treatment and control groups leads to Fisher's exact test , which is based on the hypergeometric distribution of the number of ones in the treatment group if the null hypothesis is true. When the sample sizes are large, calculating the rejection region for Fisher's Exact Test is cumbersome, but the normal approximation to the hypergeometric distribution gives an approximate test --a test whose significance level is approximately what it claims to be. In a second sampling design (the population model ), the two samples are independent random samples with replacement from two populations; conditioning on the total number of ones in the two samples again leads to Fisher's exact test, which can be approximated as before. There is another approximate approach to testing the null hypothesis in the population model: If the sample sizes are large (but the samples are drawn with replacement or are small compared to the two population sizes), the normal approximation to the distribution of the difference between the two sample percentages tends to be accurate. If the null hypothesis is true, the expected value of the difference between the sample percentages is zero, and the SE of the difference in sample percentages can be estimated by pooling the two samples. That allows one to transform the difference of sample percentages approximately into standard units , and to base an hypothesis test on the normal approximation to the probability distribution of the approximately standardized difference. Surprisingly, the resulting approximate test is essentially the normal
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Stat Hw - - Home | Text Table of Contents | Assignments |...

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