Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 88 small businesses. During a three-year period, 10 of the 71 headed by men and 3 of the 17 headed by women 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 test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative). What can we conclude (Use=0.05)?

The P-value was_________so we conclude thatThe test showed no significant difference.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 90 out of 510 businesses headed by women and 300 out of 2130 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?

The P-value was_____________so we conclude thatThe test showed strong evidence of a significant difference.

(c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a P-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses (men minus women) that fail for the settings of both (a) and (b). (Be sure to check that the conditions are met. If the conditions aren't met for one of the intervals, use the same type of interval for both)

Interval for smaller samples: _________to____________

Interval for larger samples:___________to___________