Recall that small effects may be statistically significant if the samples are la
ID: 3231871 • Letter: R
Question
Recall that small effects may be statistically significant if the samples are large. A study of small-business failures looked at 149 food-and-drink businesses. Of these, 102 were headed by men and 47 were headed by women. During a three-year period, 16 of the men's businesses and 8 of the women's businesses failed.
(a) Find the proportions of failures for businesses headed by men (sample 1) and businesses headed by women (sample 2). 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. (Round your test statistic to two decimal places and your P-value to four decimal places.)
(b) Now suppose that the same sample proportions came from a sample of 30 times as large. That is, 240 out of 1410 business headed by women and 480 out of 3060 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 it is now more significant. (Round your test statistic to two decimal places and your P-value to four decimal places.)
(c) Give the 95% confidence intervals for the difference between the proportions of men's and women's businesses that fail from Part (a) and Part (b).
(d) What is the effect of larger samples on the confidence interval?
The larger samples make the difference (and thus the length of the confidence interval) smaller.
The larger samples make the margin of error (and thus the length of the confidence interval) larger.
The larger samples make the margin of error (and thus the length of the confidence interval) smaller.
The larger samples make the difference (and thus the length of the confidence interval) larger.
pmen = pwomen=
Explanation / Answer
(a) Find the proportions of failures for businesses headed by men (sample 1) and businesses headed by women (sample 2). These sample proportions are quite close to each other.
= 8/47 = 0.170
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. (Round your test statistic to two decimal places and your P-value to four decimal places.)
Pooled estimate p = (8 + 16)/ 149 = 0.1611
standard error of sampling se0= sqrt [ p(1-p)/N] = sqrt [ 0.1611 * 0.8389/149] = 0.03
(b) Now suppose that the same sample proportions came from a sample of 30 times as large. That is, 240 out of 1410 business headed by women and 480 out of 3060 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 it is now more significant. (Round your test statistic to two decimal places and your P-value to four decimal places.)
= 240/1410 = 0.170
Pooled estimate p = (480 + 240)/ 4470 = 0.1611
standard error of sampling se0= sqrt [ p(1-p)/N] = sqrt [ 0.1611 * 0.8389/4470] = 0.0055
= 2 P(Z = 2.3636) = 0.0181
Here it is obvious that it is very very significant result and can tell that women and men failure rates are different from each other
(c) Give the 95% confidence intervals for the difference between the proportions of men's and women's businesses that fail from Part (a) and Part (b).
For part (a):
95% CI = (pw -pm) +- 1.96 * sqrt [pw (1-pw )/Nw + pm (1-pm)/Nm ]
95 % CI = (0.170 - 0.157) +- 1.96 * sqrt [0.17*0.83/47 + 0.157*0.843/102]
95 % CI = 0.013 +- 0.1285
For part (b):
95% CI = (pw -pm) +- 1.96 * sqrt [pw (1-pw )/Nw + pm (1-pm)/Nm ]
95 % CI = (0.170 - 0.157) +- 1.96 * sqrt [0.17*0.83/1410 + 0.157*0.843/3060]
95 % CI = 0.013 +- 0.0235
(d) What is the effect of larger samples on the confidence interval?
The larger samples make the difference (and thus the length of the confidence interval) smaller. NO
The larger samples make the margin of error (and thus the length of the confidence interval) larger. NO
The larger samples make the margin of error (and thus the length of the confidence interval) smaller. YES
The larger samples make the difference (and thus the length of the confidence interval) larger. NO
OPtion one is impractical as sample size doesn't effect difference. simlarly, larger sample don't decreae confidence interval range so option b and d are wrong so optio C is the only correct one.
pmen = 16/102 = 0.157 pwomen= 8/47 = 0.170
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