The average monthly Social Security benefit fora specific ye

The average monthly Social Security benefit fora specific year for retired workers was 954.90 and for disabled workers was 894.10. Researchers used data from the Social Security records to test the claim that the difference in monthly benefits between the two groups was greater than 30.00. Based on the following information, can the researchers\' claim be supported at the 0.05 level of significance? Retired: n=60, mean=960.50 population standard deviation 98.00. Disabled: n=60, mean=902.89, population standard deviation=101.00.

Solution

Let mu1 be the mean for Retired

Let mu2 be the mean for Disabled

The test hypothesis:

Ho: mu1-mu2=30 (i.e. null hypothesis)

Ha: mu1-mu2 >30 (i.e. alternative hypothesis)

The test statisitc is

Z=(xbar1-xbar2-30)/sqrt(s1^2/n1+s2^2/n2)

=(960.5-902.89-30)/sqrt(98^2/60+101^2/60)

=1.52

It is a right-tailed test.

Given a=0.05, the critical value is Z(0.05) =1.645 (from standard normal table)

The rejection region is if Z>1.645, we reject the null hypothesis.

Since Z=1.52 is less than 1.645, we do not reject the null hypothesis.

So we can not conclude that the difference in monthly benefits between the two groups was greater than 30.00