1. For a sample of 35 items from a population for which the standard deviation is σ= 20.5 , the sample mean is 458.0. At the 0.05 level of significance, test H0: μ= 450 versus H1: μ> 450 . Determine and interpret the p-value for the test.

The following null and alternative hypotheses need to be tested:

$H_0:\mu=450$

$H_1:\mu>450$

This corresponds to a right-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 right-tailed test is $z_c = 1.6449.$

The rejection region for this right-tailed test is $R = \{z: z > 1.6449\}.$

The z-statistic is computed as follows:

$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{458-450}{20.5/\sqrt{35}}\approx2.308714$

Using the P-value approach:

The p-value is $p=P(Z>2.308714)=0.010480,$ and since $p=0.010480<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is greater than $450,$ at the $\alpha = 0.05$ significance level.