Never forget that even small effects can be statistically significant if the sam
ID: 3273689 • Letter: N
Question
Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 89 small businesses. During a three-year period, 10 of the 72 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 dose 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 alpha = 0.05)? The P-value was so we conclude that The 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 heeded by women and 300 out of 2160 businesses headed by men fall. Verify that the proportions of failures are exactly the same as m part (a). Repeat the test for the new data. What can we conclude? The P-value was so we conclude that The 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 What is the effect of larger samples on the confidence interval? The confidence interval's margin of error is reduced.Explanation / Answer
H0 : p1 = p2
Ha : p1 p2
where p1 is the proportion of the failure of business headed be men.
p2 is the proportion of the failure of business headed by women.
(a)Here n1 = 72 (for men) and n2 = 17 (for women)
so p1 = 10/72 = 0.1389
so p2 =3/17 = 0.1765
Pooled estimate p= (n1p1 +n2p2)/ (n1 +n2 ) = (10+3)/ 89 = 0.1461
Standard error of the pooled estimate se0= sqrt [p (1-p) * (1/n1 + 1/n2 )]
= sqrt [0.1461 * 0.8539 * (1/72 + 1/17)] = 0.0952
Test statistic
Z = (p1 - p2 )/se0 = (0.1765 - 0.1389)/ 0.0952 = 0.395
The P - value = 2 * Pr (Z > 0.395) = 0.6928 > 0.05
that means not statistically significant.
(b) Here n1 = 2160 (for men) and n2 = 510 (for women)
so p1 = 300/2160 = 0.1389
so p2 =90/510 = 0.1765
Pooled estimate p= (n1p1 +n2p2)/ (n1 +n2 ) = (300+90)/ (510 + 2160) = 0.1461
Standard error of the pooled estimate se0= sqrt [p (1-p) * (1/n1 + 1/n2 )]
= sqrt [0.1461 * 0.8539 * (1/510 + 1/2160)] = 0.0174
Test statistic
Z = (p1 - p2 )/se0 = (0.1765 - 0.1389)/ 0.0174 = 2.161
The P - value = 2 * Pr (Z > 2.16) = 0.0307 < 0.05
that means the results are statistically significant.
(c) 95% confidence interval for smaller sample = (p1 -p2 ) +- Z95% se0 = (0.1765 - 0.1389) +- 1.96 * 0.0952
= (-0.1490, 0.2242)
95% confidence interval for bigger sample = (p1 -p2 ) +- Z95% se0 = (0.1765 - 0.1389) +- 1.96 * 0.0174
= (0.0035, 0.0717)
The largeer sample size reduce the width of confidence interval.