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Chapter 7: Q44E (page 411)

Solder-joint inspections. Current technology uses high-resolution X-rays and lasers to inspect solder-joint defects on printed circuit boards (PCBs) (Global SMT & Packaging, April 2008). A particular manufacturer of laser-based inspection equipment claims that its product can inspect, on average, at least 10 solder joints per second when the joints are spaced .1 inch apart. The equipment was tested by a potential buyer on 48 different PCBs. In each case, the equipment was operated for exactly 1 second. The number of solder joints inspected on each run follows:

The potential buyer wants to know whether the sample data refute the manufacturer’s claim. Specify the null and alternative hypotheses that the buyer should test.

Short Answer

Expert verified

a. The null and the alternative hypotheses are \({H_0}:\mu \ge 10\)and \({H_a}:\mu < 10\)

Step by step solution

01

Given information

The data of equipment tested by a potential buyer on 48 different PCBs are as follows:

02

Specifying the null hypothesis

Null hypothesis:

\({H_0}:\mu \ge 10\)

That is, the mean inspection rate is 10 soldier joints per second.

03

Specifying the alternative hypothesis

Alternative hypothesis:

\({H_a}:\mu < 10\)

That is, the mean inspection rate is at least 10 soldier joints per second.

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Most popular questions from this chapter

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.

Accidents at construction sites. In a study published in the Business & Economics Research Journal (April 2015), occupational accidents at three construction sites in Turkey were monitored. The total numbers of accidents at the three randomly selected sites were 51, 104, and 37.

Summary statistics for these three sites are:\(\bar x = 64\)and s = 35.3. Suppose an occupational safety inspector claims that the average number of occupational accidents at all Turkish construction sites is less than 70

a. Set up the null and alternative hypotheses for the test.

b. Find the rejection region for the test using\(\alpha = .01\)

c. Compute the test statistic.

d. Give the appropriate conclusion for the test.

e. What conditions are required for the test results to be valid?

Trading skills of institutional investors. Refer to The Journal of Finance (April 2011) analysis of trading skills of institutional investors, Exercise 7.36 (p. 410). Recall that the study focused on “round-trip” trades, i.e., trades in which the same stock was both bought and sold in the same quarter. In a random sample of 200 round-trip trades made by institutional investors, the sample standard deviation of the rates of return was 8.82%. One property of a consistent performance of institutional investors is a small variance in the rates of return of round-trip trades, say, a standard deviation of less than 10%.

a. Specify the null and alternative hypotheses for determining whether the population of institutional investors performs consistently.

b. Find the rejection region for the test usingα=.05

c. Interpret the value of in the words of the problem.

d. A Minitab printout of the analysis is shown (next column). Locate the test statistic andp-value on the printout.

e. Give the appropriate conclusion in the words of the problem.

f. What assumptions about the data are required for the inference to be valid?

Hotels’ use of ecolabels. Refer to the Journal of Vacation Marketing (January 2016) study of travelers’ familiarity with ecolabels used by hotels, Exercise 2.64 (p. 104). Recall that adult travelers were shown a list of 6 different ecolabels, and asked, “Suppose the response is measured on a continuous scale from 10 (not familiar at all) to 50 (very familiar).” The mean and standard deviation for the Energy Star ecolabel are 44 and 1.5, respectively. Assume the distribution of the responses is approximately normally distributed.

a. Find the probability that a response to Energy Star exceeds 43.

b. Find the probability that a response to Energy Star falls between 42 and 45. c. If you observe a response of 35 to an ecolabel, do you think it is likely that the ecolabel was Energy Star? Explain.

a. List three factors that will increase the power of a test.

b. What is the relationship between b, the probability of committing a Type II error, and the power of a test?
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