Never forget that even small effects can be statistically significant if the samples are large. Of 148 small businesses, 106 were headed by men and 42 were headed by women. During a three-year period, 15 of the men's businesses and 7 of the women's businesses failed.
a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the P-value for the z test of the hypothesis that the same proportion of women's and men's businesses fail. (use the two-sided alternative.) The test is very far from being significant.
b) Now suppose that the same sample proportions came from a sample 30 times as large. That is, 210 out of 1260 businesses headed by women and 450 out of 3180 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in (a). Repeat the z test for the new data, and show that tit is now significant at the ∝=0.05 level.
c) It is wise to use a confidence interval to estimate the size of an effect, rather than just giving a P-value. Give a 95% confidence intervals for the difference between the proportions of women's and men's businesses that fail for the settings of both (a) and (b). What is the effect of larger samples on the confidence interval?
This solution discusses underlying statistical concepts and calculates the proportions of men and women in failed businesses, applies z-tests to the data and explains the effect of a larger sample size.