Comparing Two Independent Population Proportions Quiz

PDF page 437: keeps each population from being over-sampled and causing biased results. Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance in the sampling. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the two population proportions. Like the case of differences in sample means, we construct a sampling distribution for differences in sample proportions:

PDF page 437: ⎛ ' ' ⎞ ' ' p - p where p = X A and p = X B are the sample proportions for the two sets of data in question. XA and XB B ⎝ A B⎠ A n n B A are the number of successes in each sample group respectively, and nA and nB are the respective sample sizes from the two groups. Again we go the Central Limit theorem to find the distribution of this sampling distribution for the differences in sample proportions. And again we find that this sampling distribution, like the ones past, are normally distributed as proved by the Central Limit Theorem, as seen in Figure 10.5 .

PDF page 437: ⎛ ' ' ⎞ ' ' p - p where p = X A and p = X B are the sample proportions for the two sets of data in question. XA and XB B ⎝ A B⎠ A n n B A are the number of successes in each sample group respectively, and nA and nB are the respective sample sizes from the two groups. Again we go the Central Limit theorem to find the distribution of this sampling distribution for the differences in sample proportions. And again we find that this sampling distribution, like the ones past, are normally distributed as proved by the Central Limit Theorem, as seen in Figure 10.5 .

PDF page 438: 1 1 2 0 Most common, however, is the test that the two proportions are the same. That is, H : p = p

PDF page 438: B A 0 Ha : p ≠ p B A To conduct the test, we use a pooled proportion, pc. The pooled proportion is calculated as follows: x + x

PDF page 438: Generally, the null hypothesis allows for the test of a difference of a particular value, δ0, just as we did for the case of differences in means. H : p − p = δ

PDF page 439: The test statistic (z-score) is: (p′ − p′ ) − δ

PDF page 439: A bank has recently acquired a new branch and thus has customers in this new territory. They are interested in the default rate in their new territory. They wish to test the hypothesis that the default rate is different from their current customer base. They sample 200 files in area A, their current customers, and find that 20 have defaulted. In area B, the new customers, another sample of 200 files shows 12 have defaulted on their loans. At a 10% level of significance can we say that the default rates are the same or different?

PDF page 439: A bank has recently acquired a new branch and thus has customers in this new territory. They are interested in the default rate in their new territory. They wish to test the hypothesis that the default rate is different from their current customer base. They sample 200 files in area A, their current customers, and find that 20 have defaulted. In area B, the new customers, another sample of 200 files shows 12 have defaulted on their loans. At a 10% level of significance can we say that the default rates are the same or different?

PDF page 439: Solution 10.6 This is a test of proportions. We know this because the underlying random variable is binary, default or not default. Further, we know it is a test of differences in proportions because we have two sample groups, the current customer base and the newly acquired customer base. Let A and B be the subscripts for the two customer groups. Then pA and pB are the two population proportions we wish to test. Random Variable:

PDF page 439: B A 0 Ha : p ≠ p B A The words "is a difference" tell you the test is two-tailed. Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal: x + x 20 + 12

PDF page 439: Estimated proportion for group A: p′ = = = 0.1