Never forget that even small effects can be statistically significant if the sam
ID: 3062969 • 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 131 small businesses. During a three-year period, 12 of the 86 headed by men and 8 of the 45 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 that
Choose a conclusion. The test showed strong evidence of a significant difference. The test showed no significant difference.
(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 240 out of 1350 businesses headed by women and 360 out of 2580 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. 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)
Interval for smaller samples: to
Interval for larger samples: to
What is the effect of larger samples on the confidence interval?
Explanation / Answer
(a) Here i will denote men by 1 and women by 2.
so Here
p^1 = 12/86 = 0.1395
p^2 = 8/45 = 0.1778
Pooled estimate p =(12 + 8)/ (86 + 45) = 0.1527
standard error of difference = sqrt [p * (1-p) * (1/n1 + 1/n2)] = sqrt [0.1527 * (1 - 0.1527) * (1/86 + 1/45)] = 0.0662
Here Test statistic
Z = (p^1 - p^2)/se0 = (0.1778 - 0.1395)/0.0662 = 0.5785
P - value = 2 * Pr(Z > 0.5785) = 0.5633
so we show that there is no strong evidence that men and women are different in failure rate of their corporation.
(b)
so Here
p^1 = 360/2580 = 0.1395
p^2 = 240/1350 = 0.1778
Pooled estimate p =(360 + 240)/ (2580 + 1350) = 0.1527
standard error of difference = sqrt [p * (1-p) * (1/n1 + 1/n2)] = sqrt [0.1527 * (1 - 0.1527) * (1/2580 + 1/1350)] = 0.0121
Here Test statistic
Z = (p^1 - p^2)/se0 = (0.1778 - 0.1395)/0.0121 = 3.1654
P - value = 2 * Pr(Z > 3.1654) = 0.0015
So here we can say that there is strong evidence that failure rate for women is different than men.
(c) Here 95% confidence interval in case 1 =(p^1 - p^2) +- Z95% se0 = (0.1778 - 0.1395) +- 1.96 * 0.0662
= (-0.0915, 0.1679)
95% confidence interval in case 2 =(p^1 - p^2) +- Z95% se0 = (0.1778 - 0.1395) +- 1.96 * 0.0121
= (0.0146, 0.0619)
so here the larger samples have smallr confidence intervals than the smaller samples.