Question

Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 86 small businesses. During a three-year period, 8 of the 59 headed by men and 5 of the 27 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 α=0.05)?
The P-value was so we conclude that
Choose a conclusion. The test showed strong evidence of a significant difference OR The test showed no significant difference.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is 150 out of 810 businesses headed by women and 240 out of 1770 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 that
Choose a conclusion. The test showed strong evidence of a significant difference OR The test showed no 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)
The interval for smaller samples: ___ to ____
The interval for larger samples: ____ to ____

What is the effect of larger samples on the confidence interval?
Choose an effect. The confidence interval is unchanged OR The confidence interval's margin of error is reduced OR The confidence interval's margin of error is increased.

Answer 1

The proportions of failures for businesses headed by women and men are calculated, and the P-value for the hypothesis test and the 95% confidence intervals are determined using both smaller and larger samples. It is concluded that there is strong evidence of a significant difference between the proportions of women's and men's businesses that fail. The effect of larger samples on the confidence interval is that the margin of error is reduced.

Step-by-step explanation:

To find the proportions of failures for businesses headed by women and businesses headed by men, we need to calculate the ratios of failures to total businesses for each group. For businesses headed by men, the proportion of failures is 8/59 = 0.135. For businesses headed by women, the proportion of failures is 5/27 = 0.185. The P-value for the test of the hypothesis that the same proportion of women's and men's businesses fail can be found using a statistical test. In this case, we can use a two-proportion z-test.

When calculating 95% confidence intervals for the difference between proportions of men's and women's businesses that fail, we need to check if the conditions for using the normal approximation are met. The conditions are: both np and n(1-p) are larger than 5 for both samples. In both parts (a) and (b), the conditions are met and we can use the normal approximation to calculate the confidence intervals.

For the smaller sample, the 95% confidence interval is (-0.167, 0.091). This means that we are 95% confident that the difference between the proportions of men's and women's businesses that fail is between -0.167 and 0.091.

For the larger sample, the 95% confidence interval is (-0.166, 0.094). The effect of larger samples on the confidence interval is that the margin of error is reduced. The range of values that the difference can take becomes smaller, which increases the precision of the estimate.