Answer to Question #342460 in Statistics and Probability for erika miguel

Question

Given the following information, construct the rejection region. Show the solution in

a step-by-step procedure.

H 0 : = 45

H a : < 45

= 40, = 12, n = 32, = 0.01

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=45$

$H_a:\mu\not=45$

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

Based on the information provided, the significance level is $\alpha = 0.01,$ $df=n-1=31$ degrees of freedom, and the critical value for a two-tailed test is $t_c =2.744042.$The rejection region for this two-tailed test is $R = \{t:|t|> 2.744042\}.$

The t-statistic is computed as follows:

$$t=\dfrac{40-45}{12/\sqrt{32}}=-2.357$$

Since it is observed that $|t| = 2.357<2.744042=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for two-tailed $df=31$ degrees of freedom, $t=-2.357$ is $p=0.024919,$ and since $p=0.024919>0.01=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$ is different than 45, at the $\alpha = 0.01$ significance level.