Question

This question is based on Ch10, but we can solve it using our knowledge from Ch9. In Ch 8, we created confidence intervals to test whether the means differed in a statistically significant manner between two independent (unrelated) populations, like males and females. The sample point estimator in the confidence interval was the difference in the sample means between the 2 samples (xbar - ybar). We created a confidence interval of the form:
(xbar-ybar) +/- (Zα/2)[standard error of (xbar-ybar)]
We can use the confidence interval to test the null hypothesis that the population means are the same or not (Ho: μx – μy = 0).  If the confidence interval did not contain 0, then it was unlikely that the true difference in means is zero, and we concluded that there is probably a real difference in the population means

An equivalent hypothesis test in Ch 10 tests whether 2 means are different using the same sample estimator (xbar - ybar) and our knowledge of the sampling distribution for this estimator for the difference in means.

We test the null hypothesis Ho: μx – μy = 0
against an alternative H1: μx – μy ≠ 0

Using the test statistic Z = [(xbar - ybar) - hypothesized difference] / [std error of (xbar-ybar)]
Z = [(xbar - ybar) - 0] / [std error of (xbar-ybar)]

The critical value for the 2-sided H1 is Zα/2, which equals 1.96 for a test with level of significance α=5%.

Question: In 2012, the sample average mean hourly earnings for males 25-34 in the US was $25.30 and for females it was $21.50. The sample size of males was nx=2004 and the sample Sx=$12.09. The sample size of females was ny=1951 and Sy = $9.99.

If the standard error for the difference in sample means is (12.09²/2004 + 9.99²/1951) ^1/2 = 0.35, test the null hypothesis that mean wages do not differ significantly between males and females against the alternative that they do at α=1%.

• A.

  The test statistic = 2.58; the p-value is 0.005, and there is NO evidence that mean wages differ between males and females.

• B.

  The test statistic = 10.86; the p-value is 0, and there is NO evidence that mean wages differ between males and females.

• C.

  The test statistic = 10.86; the critical value is 2.58, and there is strong evidence that mean wages differ between males and females.

• D.

  The test statistic = 2.58; the p-value is 0.005, and there is strong evidence that mean wages differ between males and females.

Answer 1