Question

Test the Hypothesis If the Mean travel time in minutes between Point A to Point B is equal to the mean of the travel time in minutes from Point B to your A. First you must find the mean and standard deviations. Then perform and list the complete required steps for the TWO required Hypothesis tests and ALSO USE THE P-VALUE AS A REJECTION RULE FOR BOTH TESTS. One Hypothesis test is an F test for the equality of the variances of travel Times and the second test is a T test for the equality of the means of travel times in minutes. The F test must be performed first in order to select either Case1 or Case 2 for the T-test. PLEASE SHOW HOW YOU OBTAINED ALL ANSWERS

Recorded Time values in minutes from point A to point B: 32, 34, 51, 30, 29, 35, 36, 29, 32, 29, 33, 32, 29, 30, 33, 30, 30, 33, 30, 31, 35, 35, 34, 32, 33, 33, 31, 33, 34, 30, 30, 29, 34, 32, 36, 29, 30, 32, 30, 33, 31

n1=41

Recorded Time values in minutes from point B to point A: 36, 28, 48, 28, 27, 54, 34, 29, 26, 34, 33, 42, 29, 34, 31, 48, 27, 42, 28, 45, 26, 43, 32, 41, 30, 36, 27, 44, 29, 29, 35, 26, 31, 28, 27, 28, 32, 41, 34, 28, 31

n2=41

Answer 1

The provided sample means are shown below:

and

Also, the provided sample standard deviations are:

and the sample sizes are and

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha:

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

Testing for Equality of Variances

A F-test is used to test for the equality of variances. The following F-ratio is obtained:

The critical values are and , and since , then the null hypothesis of equal variances is rejected.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=80.

Hence, it is found that the critical value for this two-tailed test is tc​=1.99, for α=0.05 and df = 80

The rejection region for this two-tailed test is

(3) Test Statistics

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.2752, and since , it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.05 significance level.

Graphically