Question

Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below:

Population 1: 64, 65, 63, 69, 70, 65, 68

Population 2: 69, 77, 73, 71, 79, 69, 70, 71

Is there evidence, at an α=0.001α=0.001 level of significance, to conclude that there those who exercise regularly have lower resting heart rates? (Use the conservative method for computing degrees of freedom). Carry out an appropriate hypothesis test, filling in the information requested.

A. The value of the standardized test statistic:

B. The p-value is

C. Your decision for the hypothesis test:

A. Reject H0H0.
B. Reject H1H1.
C. Do Not Reject H0H0.
D. Do Not Reject H1H1.

Answer 1

The random sample is given for the two groups, namely as Population1 (Group which Exercises regularly) and Population2 (DO not exercise regularly). Now the hypothesis for to test the claim that the group which exercises regularly(Population1) have lower resting heart rates than that the group that do not exercise regularly.

Hypothesis:

Difference between the true mean resting heart rates for the population is not different     than zero. Or we can say that the there is no difference in the mean resting heart rates for the two groups.

The difference of the true mean resting heart rates for the two group is less than zero, which means the group(Exercises regularly) have lower resting heart rates than the group who does not exercise regularly.

The test we are using here is Two-sample t-test with EQUAL VARIANCE.

The formula for the test statistic is:   

; with degrees of freedom, df=n₁+n₂-2

   ;   s_p² is known as the pooled variance

(A) Calculation for test-statistic:

Group1 (Exercises Regularly):

Random sample for Group 1 is : {64, 65, 63, 69, 70, 65, 68}

The sample mean and sample variance for the group1(Exercises regularly) is:

Group2 (Do not exercise regularly):

The sample mean and sample variance for the group2(Do not Exercises regularly) is:

Calculation for pooled variance

Calculation for the degrees of freedom(df):

And the other way to find the df is:

Calculation for test-statistic:

and

                                                     

The test statistic is calculated as t=-3.569

(B) P-value:

(C)

Since,

In other words, at we FTR(Fail to reject) H_o , and conclude that the data does not provide enough evidence to support alternative hypothesis, i.e., . So we can say that the true mean resting heart rates for the population\group who do exercise regularly and the population\group who do not exercise regularly are not significantly different.