Question

A quality-assurance inspector periodically exam-ines the output of a machine to determine whether it is properly adjusted. When set properly, the machine pro-duces nails having a mean length of 2.000 inches, with a standard deviation of 0.070 inches. For a sample of 35 nails, the mean length is 2.025 inches. Using the 0.01 level of significance, examine the null hypothesis that the machine is adjusted properly. Determine and interpret the p-value for the test.

Answer 1

Answer:

Step-by-step explanation:

From the given information;

The null hypothesis & alternative hypothesis:

$\mathbf{H_o:\mu = 2} \\ \\ \mathbf{H_1 : \mu \ne 2}$

The test statistics can be computed as:

$Z = \dfrac{\overline x - \mu}{\dfrac{\sigma}{\sqrt{n}}}$

$Z = \dfrac{2.025- 2}{\dfrac{0.07}{\sqrt{35}}}$

$Z = \dfrac{0.025}{\dfrac{0.07}{\sqrt{35}}}$

$Z =2.11$

The p-value = 2(Z > 2.11)  since this is a two-tailed test

The p-value = 2( 1 - Z < 2.11)

The p-value = 2 (1 -0.9826)

The p-value = 2 (0.0174)

The p-value = 0.0348

Decision Rule: To reject the $\mathbf{H_o}$  if the p-value is less than $\mathbf{H_o}$

Conclusion: We fail to reject $\mathbf{H_o}$ and conclude that the population mean = 2, thus the machine is properly adjusted.