Question

1. Native American [8.50, 9.48, 8.65]

2. Caucasian [8.27, 8.20, 8.25, 8.14]

alpha = 0.10

Test the difference between the two groups using a two sample t-test, using p value approach, with detailed formulas and conclude

Answer 1

1 represents Native American and 2 represents Caucasian.

The sample means are shown below:

Also, the sample standard deviations are:

and the sample sizes are n1​=3 and n2​=3.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ ≠ μ2​

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 FL​=0.053 and FU​=19, and since F = 82.762, then the null hypothesis of equal variances is rejected.

(2) Rejection Region

The significance level is α = 0.10, and the degrees of freedom are df = 2.048. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal:

(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

The p-value is 2* P(|t| > 2.158 ) with 2.048 degree of freedom is 0.1606.

Using the P-value approach: The p-value is p = 0.1606, and since p = 0.1606 ≥ 0.10, 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 0.10 significance level.