Question

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

a step-by-step procedure.

1. H 0 : = 84

H a : 84

m= 87, s= 10, n = 35, $\alpha$ = 0.05

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=84$

$H_1:\mu\not=84$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a two-tailed test is $z_c = 1.96.$

The rejection region for this two-tailed test is $R = \{z:|z|> 1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{87-84}{10/\sqrt{35}}\approx1.7748$$

Since it is observed that $z=1.7748<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(z>1.7748)=0.075931,$ and since $p=0.075931>0.05=\alpha,$ 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 84, at the $\alpha = 0.05$ significance level.