Question

Preliminary data analyses indicate that you can consider the assumptions for using nonpooled​ t-procedures satisfied. Researchers randomly and independently selected 30 prisoners diagnosed with chronic posttraumatic stress disorder​ (PTSD) and 24 prisoners that were diagnosed with PTSD but had since recovered​ (remitted). The​ ages, in​ years, at arrest yielded the summary statistics shown in the table to the right. At the 10​% significance​ level, do the data provide sufficient evidence to conclude that a difference exists in the mean age at arrest of prisoners with chronic PTSD and remitted​ PTSD? Chronic Remitted x overbar 1 equals 26.9 x overbar 2 equals 22.4 s 1 equals 4 s 2 equals 8 n 1 equals 30 n 2 equals 24 What are the hypotheses for the nonpooled​ t-test? A. H0​: mu1equalsmu2 Ha​: mu1greater thanmu2 B. H0​: mu1greater than or equalsmu2 Ha​: mu1less thanmu2 C. H0​: mu1equalsmu2 Ha​: mu1not equalsmu2 D. H0​: mu1equalsmu2 Ha​: mu1less thanmu2 Find the test statistic. tequals nothing ​(Round to three decimal places as​ needed.) Find the​ P-value. Pequals nothing ​(Round to four decimal places as​ needed.) What is the conclusion of the hypothesis​ test? ▼ the null​ hypothesis, meaning that the data ▼ sufficient evidence to conclude that a difference exists in the mean age at arrest of prisoners with chronic PTSD and remitted PTSD.

Answer 1

The null and alternative hypothesis for the non-pooled t-test is:

i.e., the mean age at arrest of prisoners with chronic PTSD is not different than the mean age at arrest of prisoners with remitted PSTD.

i.e., there is a difference in the mean age at arrest of prisoners with chronic PTSD and remitted​ PTSD.

We need to test this hypothesis with a given significance level of

The following data is given-

	Chronic PSTD	Remitted PSTD
Sample mean
Sample standard deviation
Sample size

Test-statistic:             

Degrees of freedom:  

Calculation for test-statistic-

So, the test statistic is calculated as

P-value: test-statistic is and , and since we are testing a two-tailed hypothesis, so the p-value is calculated as-

So, the p-value for the sample test-statistic is calculated as

Conclusion: Now we compare the p-value=0.0193 with the significance level , to make a decision that whether we should reject null hypothesis or do not reject null hypothesis.

Since,

So, at significance level of the sample data provides sufficient evidence to reject null hypothesis H₀.

In other words, "We reject the null hypothesis H₀, meaning that the data provides sufficient evidence to conclude that a difference exists in the mean age at arrest of the prisoners with chronic PSTD and remitted PSTD. "