Question

Medicare spending per patient in different U.S. metropolitan areas may differ. Based on the sample data below, answer the questions that follow to determine whether the average spending in the northern region is significantly less than the average spending in the southern region at the 1 percent level.
   

Medicare Spending per Patient (adjusted for age, sex, and race)
Statistic	Northern Region			Southern Region
Sample mean	$	3,123		$	8,456
Sample standard deviation	$	1,546		$	3,678
Sample size		14	patients		16	patients

Answer 1

Let be the population mean for northern region and be the population mean for southern region

Here the hypothesis are as follow

Null hypothesis, Ho;

Alternate hypothesis,Ha;

This corresponds to a left-tailed test

Based on the information provided, the significance level is α=0.01, and the degrees of freedom are df=n1+n2-2= 16+14-2= 28..

Hence, it is found that the critical value for this left-tailed test is tc​=−2.467 (from t-table at  α=0.01 and df=28)

The rejection region for this left-tailed test is R={t:t<−2.467}.

The formula for compting test statistic is

Since it is observed that t = -5.045 & t=−5.045<tc​=−2.467, it is then concluded that the null hypothesis is rejected.

Hence there is enough evidence to claim that the average spending in the northern region is significantly less than the average spending in the southern region at the 1 percent level.