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Chapter 7: Q37E (page 410)

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

Expert verified

The null and the alternative hypothesis are

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

Step by step solution

01

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.

02

 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.

03

 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.

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Most popular questions from this chapter

Customers who participate in a store鈥檚 free loyalty card program save money on their purchases but allow the store to keep track of the customer鈥檚 shopping habits and potentially sell these data to third parties. A Pew Internet & American Life Project Survey (January 2016) revealed that 225 of a random sample of 250 U.S. adults would agree to participate in a store loyalty card program, despite the potential for information sharing. Letp represent the true proportion of all customers who would participate in a store loyalty card program.

a. Compute a point estimate ofp

b. Consider a store owner who claims that more than 80% of all customers would participate in a loyalty card program. Set up the null and alternative hypotheses for testing whether the true proportion of all customers who would participate in a store loyalty card program exceeds .8

c. Compute the test statistic for part b.

d. Find the rejection region for the test if 伪=.01.

e. Find the p-value for the test.

f. Make the appropriate conclusion using the rejection region.

g. Make the appropriate conclusion using the p-value.

For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region.

a. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.1\)

b. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.05\)

c. \({H_0}:\mu \ge {\mu _0}\) and \({H_a}:\mu < {\mu _0};\alpha = 0.01\)

d. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.05\)

e. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.1\)

f. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.01\)

g. For each rejection region specified in parts a鈥揻, state the probability notation in z and its respective Type I error value.

Packaging of a children鈥檚 health food. Can packaging of a healthy food product influence children鈥檚 desire to consume the product? This was the question of interest in an article published in the Journal of Consumer Behaviour (Vol. 10, 2011). A fictitious brand of a healthy food product鈥攕liced apples鈥攚as packaged to appeal to children (a smiling cartoon apple was on the front of the package). The researchers showed the packaging to a sample of 408 school children and asked each whether he or she was willing to eat the product. Willingness to eat was measured on a 5-point scale, with 1 = 鈥渘ot willing at all鈥� and 5 = 鈥渧ery willing.鈥� The data are summarized as follows: \(\bar x = 3.69\) , s = 2.44. Suppose the researchers knew that the mean willingness to eat an actual brand of sliced apples (which is not packaged for children) is \(\mu = 3\).

a. Conduct a test to determine whether the true mean willingness to eat the brand of sliced apples packaged for children exceeded 3. Use\(\alpha = 0.05\)

to make your conclusion.

b. The data (willingness to eat values) are not normally distributed. How does this impact (if at all) the validity of your conclusion in part a? Explain.

A random sample of 41 observations from a normal population possessed a mean \(\bar x = 88\) and a standard deviation s = 6.9.

a. Test \({H_0}:{\sigma ^2} = 30\) against \({H_a}:{\sigma ^2} > 30\). Use\(\alpha = 0.05.\)

Specify the differences between a large-sample and a small-sample test of a hypothesis about a population mean m. Focus on the assumptions and test statistics.
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