Question

Producer's and consumer's risk. In quality-control applications of hypothesis testing, the null and alternative hypotheses are frequently specified as\({H_0}\)The production process is performing satisfactorily. \({H_a}\): The process is performing in an unsatisfactory manner. Accordingly, \(\alpha \) is sometimes referred to as the producer's risk, while \(\beta \)is called the consumer's risk (Stevenson, Operations Management, 2014). An injection molder produces plastic golf tees. The process is designed to produce tees with a mean weight of .250 ounce. To investigate whether the injection molder is operating satisfactorily 40 tees were randomly sampled from the last hour's production. Their weights (in ounces) are listed in the following table.

a. Write \({H_0}\) and \({H_a}\) in terms of the true mean weight of the golf tees, \(\mu \).

b. Access the data and find \(\overline x \)and s.

c. Calculate the test statistic.

d. Find the p-value for the test.

e. Locate the rejection region for the test using\({H_a} = 0.01\).

f. Do the data provide sufficient evidence to conclude that the process is not operating satisfactorily?

g. In the context of this problem, explain why it makes sense to call \(\alpha \)the producer's risk and \(\beta \)the consumer's risk.

Short answer

The null and the alternative hypothesis are

\({H_0}:\mu = 0.25\) and \({H_0}:\mu \ne 0.25\)

Step-by-step solution

Given information.

In quality-control applications of hypothesis testing, the null and alternative hypotheses are frequently specified as

\({H_0}\): The production process is performing satisfactorily.

\({H_a}\): The process is performing in an unsatisfactory manner.

 Concept of the null and the alternative hypothesis

The null hypothesis of a test always predicts no effect or no relationship between variables, whereas the alternative hypothesis outlines your study's prediction of an effect or relationship.

 Setting up the null and the alternative hypothesis.

a.

Null hypothesis:

\({H_0}:\mu = 0.25\)

The true mean weight of the golf tees is 0.25 ounces.

Alternative hypothesis:

\({H_0}:\mu \ne 0.25\)

The true mean weight of the golf tees is not equal to 0.25 ounces.