Question

Is College Worth It? In the New York Times article "College Graduates Fare Well in Jobs Market, Even Through Recession," C. Rampell noted that college graduates have suffered through the recession and lackluster recovery with remarkable resilience. Of a random sample of 1020 college graduates, 35 were unemployed; and of a random sample of 1008 high-school graduates (no college), 69 were unemployed.

a. At the 1 T significance level, do the data provide sufficient evidence to conclude that college graduates have a lower unemployment rate than high-school graduates?

b. Find and interpret a$98\%$ confidence interval for the difference in unemployment rates of college and high-school graduates.

Short answer

a) The value of the test statistic is -$3.48$ and P-value is$0.000$

b) The $98\%$confidence interval is $(-0.0569,-0.0114)$

Step-by-step solution

Part(a) Step 1: Given Information

For a random sample of $1020$college graduates, 35 were unemployed.

Part(a) Step 2: Explanation

Hypothesis test:

States the null and alternative hypothesis:

Null hypothesis:

${\mathrm{H}}_0:p_1=p_2$

College graduates do not have a lower unemployment rate than high-school graduates.

Alternative hypothesis:

${\mathrm{H}}_0:p_1<p_2$

College graduates have a lower unemployment rate than high-school graduates.

Calculated the test statistic and $P-$value by MINITAB

MINITAB Output

The value of test statistic is$-3.48$and $P$-value is $0.000.$

If $P\le \alpha$, then reject the null hypothesis.

Conclusions

Significant level is $\alpha =0.01$

Here, the $P$- value is 0 is lesser than the level of significance is

$P(=0.000)<\alpha (=0.01)$

Therefore, by the rejection rule, it can be concluded that there is evidence to reject the null hypothesis ( $\left.H_0\right)$at $\alpha =0.01$

Therefore, the data provide sufficient evidence to conclude that college graduates have a lower unemployment rate than high-school graduates.

Part(b) Step 1: Given Information

For a random sample of $1020$ college graduates, 35 were unemployed.

Part(b) Step 2: Explanation

Confidence interval:

Compute confidence interval by MINITAB

MINITAB output

The$98\%$ confidence interval is $(-0.0569,-0.0114)$

There is $98\%$ confidence that the difference in unemployment rates of college and high-school graduates lies between $-0.0569$ and $-0.0114.$