Question

The null and alternate hypotheses are:

H_o: m₁ = m₂     H₁: m₁ ≠ m₂

A random sample of 15 observations from the first population revealed a sample mean of 350 and a sample standard deviation of 12. A random sample of 17 observations from the second population revealed a sample mean of 342 and a sample standard deviation of 15. At the .10 significance level, is there a difference in the population means?

1. Is this a one-tailed or a tow-tailed test?
2. State the decision rule.
3. Compute the value of the test statisti
4. What is your decision regarding Ho?
5. What is the p-value?

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are n1 = 15 and n2 = 17.

(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.

(2) Rejection Region

Based on the information provided, the significance level is α = 0.1, and the degrees of freedom are df = 30. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this two-tailed test is t_c = 1.697,α = 0.1 and df = 30.

(3) Test Statistics

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

t = 1.651

(4) Decision about the null hypothesis

Since it is observed that |t| = 1.651< t_c = 1.697, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.1093, and since p = 0.1093 > 0.1, 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 0.1 significance level.