Question

Consider the following hypothesis test.

H₀: μ ≥ 40

H_a: μ < 40

A sample of 36 is used. Identify the p-value and state your conclusion for each of the following sample results. Use

α = 0.01.

(a)

x = 39 and s = 5.3

*Find the value of the test statistic. (Round your answer to three decimal places.)

* Find the p-value. (Round your answer to four decimal places.)

p-value =

*State your conclusion:

Reject H₀. There is sufficient evidence to conclude that μ < 40.

Do not reject H₀. There is sufficient evidence to conclude that μ < 40.    

Do not reject H₀. There is insufficient evidence to conclude that μ < 40.

Reject H₀. There is insufficient evidence to conclude that μ < 40.

Answer 1

The provided sample mean is 39 and the sample standard deviation is 5.3, and the sample size is 36.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ ≥ 40

Ha: μ < 40

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a left-tailed test is t_c = -2.438

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that t = -1.132 > t_c = -2.438 , it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.1326 , and since p = 0.1326 > 0.01 , 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 μ is less than 40, at the 0.01 significance level.

Do not reject H₀. There is insufficient evidence to conclude that μ < 40.