Answer to Question #195984 in Statistics and Probability for zub

Question

A high-end computer manufacturer sets the retail cost of their computers based on the manufacturing cost. The average manufacturing cost of their computers is reported to be $1800 with a standard deviation of $500. However, the executive of the company suspected that there might be an error in the reporting. The company randomly selects 35 computers from its facilities and finds that the mean cost to produce each computer is $1950.

a) State the appropriate hypotheses to test whether the mean manufacturing cost is the same as the one reported by the manufacturer. [2 marks]

b) Describe Type I and Type II errors in this context, and comment on the severity of each type of error. [3 marks]

c) Test the hypotheses stated in part (a) at a 1% level of significance, and comment on your findings. [5 marks]

Answer 1

a)

$H_0:$ the mean manufacturing cost is the same as the one reported by the manufacturer.

$H_{\alpha}:$ the mean manufacturing cost is not the same as the one reported by the manufacturer.

c)

$z=\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}=\frac{1950-1800}{500/\sqrt{35}}=1.77$

Since z-value is less than critical value $Z_{0.99}=2.58$ , we can accept null hypothesis $H_0$ , that the mean manufacturing cost is the same as the one reported by the manufacturer.

b)

Type I error is the rejection of a true null hypothesis as the result of a test procedure.  In this context, this is the rejection of hypothesis, that the mean manufacturing cost is the same as the one reported by the manufacture at a 1% level of significance.

Type II error is the failure to reject a false null hypothesis as the result of a test procedure. In this context, this is the failure to reject of hypothesis, that the mean manufacturing cost is not the same as the one reported by the manufacture at a 1% level of significance.