Never forget that even small effects canbe statistically significant if the samples are

Chapter 0, Problem 21.24

(choose chapter or problem)

Never forget that even small effects canbe statistically significant if the samples are large. To illustrate this fact, return tothe study of 148 small businesses described in Exercise 20.37 (page 510).Of these, 106 were headed by men and 42 were headed by women. During athree-year period, 15 of the mens businesses and 7 of the womens businessesfailed.(a) Find the proportions of failures for businesses headed by women andbusinesses 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 ofwomens and mens businesses fail. (Use the two-sided alternative.) The test isvery far from being significant.(b) Now suppose that the same sample proportions came from a sample 30 timesas large. That is, 210 out of 1260 businesses headed by women and 450 out of3180 businesses headed by men fail. Verify that the proportions of failures areexactly the same as in (a). Repeat the z test for the new data, and show that it isnow significant at the = 0.05 level.(c) It is wise to use a confidence interval to estimate the size of an effect, ratherthan just giving a P-value. Give 95% confidence intervals for the differencebetween the proportions of womens and mens businesses that fail for the settingsof both (a) and (b). What is the effect of larger samples on the confidenceinterval?