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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

Accuracy of price scanners at Walmart. Refer to Exercise 6.129 (p. 377) and the study of the accuracy of checkout scanners at Walmart stores in California. Recall that the National Institute for Standards and Technology (NIST) mandates that for every 100 items scanned through the electronic checkout scanner at a retail store, no more than two should have an inaccurate price. A study of random items purchased at California Walmart stores found that 8.3% had the wrong price (Tampa Tribune, Nov. 22, 2005). Assume that the study included 1,000 randomly selected items.

a. Identify the population parameter of interest in the study.

b. Set up H0 and Ha for a test to determine if the true proportion of items scanned at California Walmart stores exceeds the 2% NIST standard.

c. Find the test statistic and rejection region (at a=0.05 ) for the test.

d. Give a practical interpretation of the test.

e. What conditions are required for the inference, part d, to be valid? Are these conditions met?

Salaries of postgraduates. The Economics of Education Review (Vol. 21, 2002) published a paper on the relationship between education level and earnings. The data for the research was obtained from the National Adult Literacy Survey of more than 25,000 respondents. The survey revealed that males with a postgraduate degree have a mean salary of \(61,340 (with standard error \(Sx\) = \)2,185), while females with a postgraduate degree have a mean of \(32,227 (with standard error \(Sx\) = \)932).

  1. The article reports that a 95% confidence interval for \[{\bf{\mu }}M\] , the population mean salary of all males with post-graduate degrees, is (\(57,050, \)65,631). Based on this interval, is there evidence to say that \[{\bf{\mu }}M\] differs from \(60,000? Explain.
  2. Use the summary information to test the hypothesis that the true mean salary of males with postgraduate degrees differs from \)60,000. Use \(\alpha \) =.05.
  3. Explain why the inferences in parts a and b agree.
  4. The article reports that a 95% confidence interval for \(\mu F\) , the population mean salary of all females with post-graduate degrees, is (\(30,396, \)34,058). Based on this interval, is there evidence to say that \(\mu F\)differs from \(33,000? Explain.
  5. Use the summary information to test the hypothesis that the true mean salary of females with postgraduate degrees differs from \)33,000. Use \(\alpha \) =.05.
  6. Explain why the inferences in parts d and e agree.

-Question:Consumers’ use of discount coupons. In 1894, druggist Asa Candler began distributing handwritten tickets to his customers for free glasses of Coca-Cola at his soda fountain. That was the genesis of the discount coupon. In 1975, it was estimated that 65% of U.S. consumers regularly used discount coupons when shopping. In a more recent consumer survey, 81% said they regularly redeem coupons (NCH Marketing Services 2015 Consumer Survey). Assume the recent survey consisted of a random sample of 1,000 shoppers.

a. Does the survey provide sufficient evidence that the percentage of shoppers using cents-off coupons exceeds 65%? Test using α = 0.05.

b. Is the sample size large enough to use the inferential procedures presented in this section? Explain.

c. Find the observed significance level for the test you conducted in part a and interpret its value.

Improving the productivity of chickens. Refer to the Applied Animal Behaviour Science (October 2000) study of the color of string preferred by pecking domestic chickens, Exercise 6.124 (p. 376). Recall that n = 72 chickens were exposed to blue string and the number of pecks each chicken took at the string over a specified time interval had a mean of\(\overline x = 1.13\,\)pecks and a standard deviation of s = 2.21 pecks. Also recall that previous research had shown that m = 7.5 pecks if chickens are exposed to white string.

a. Conduct a test (at\(\alpha = 0.01\)) to determine if the true mean number of pecks at blue string is less than\(\mu = 7.5\)pecks.
b. In Exercise 6.122, you used a 99% confidence interval as evidence that chickens are more apt to peck at white string than blue string. Do the test results, part a, support this conclusion? Explain

Oxygen bubble velocity in a purification process. Refer to the Chemical Engineering Research and Design (March 2013) study of a method of purifying nuclear fuel waste, Exercise 6.35 (p. 349). Recall that the process involves oxidation in molten salt and tends to produce oxygen bubbles with a rising velocity. To monitor the process, the researchers collected data on bubble velocity (measured in meters per second) for a random sample of 25 photographic bubble images. These data (simulated) are reproduced in the accompanying table. When oxygen is inserted into the molten salt at a rate (called the sparging rate) of \(3.33 \times {10^{ - 6}}\) , the researchers discovered that the true mean bubble rising velocity \(\mu = .338\)

a. Conduct a test of hypothesis to determine if the true mean bubble rising velocity for the population from which the sample is selected is\(\mu = .338\)Use\(\alpha = .10\).

0.275 0.261 0.209 0.266 0.265 0.312 0.285 0.317 0.229 0.251 0.256 0.339 0.213 0.178 0.217 0.307 0.264 0.319 0.298 0.169 0.342 0.270 0.262 0.228 0.22
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