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Chapter 7: Q137SE (page 444)

EPA limits on vinyl chloride. The EPA sets an airborne limit of 5 parts per million (ppm) on vinyl chloride, a colorless gas used to make plastics, adhesives, and other chemicals. It is both a carcinogen and a mutagen (New Jersey Department of Health, Hazardous Substance Fact Sheet, 2010). A major plastics manufacturer, attempting to control the amount of vinyl chloride its workers are exposed to, has given instructions to halt production if the mean amount of vinyl chloride in the air exceeds 3.0 ppm. A random sample of 50 air specimens produced the following statistics: \(\overline x = 3.1\)ppm,\(s = 0.5\)ppm.

a. Do these statistics provide sufficient evidence to halt the production process? Use\(\alpha = 0.01\).

b. If you were the plant manager, would you want to use a large or a small value for\(\alpha \)the test in part a? Explain.

c. Find the p-value for the test and interpret its value

Short Answer

Expert verified

The null and the alternative hypotheses are \({H_0}:\mu = 3.0\)and \({H_a}:\mu > 3.0\)

Step by step solution

01

Given information

The information is regarding a significant plastics manufacturer attempting to control the amount of vinyl chloride its workers are exposed.

A random sample of 50 air specimens produced the following statistics

\(\overline x = 3.1\)ppm

\(s = 0.5\)ppm

02

Concept of the null and the alternative hypothesis

The alternative hypothesis of a test expresses your research’s prediction of an effect or relationship. In contrast, the null hypothesis of a test always predicts no effect or no association between variables.

03

Setting up the null and the alternative hypothesis

a.

The claim is that the mean amount of vinyl chloride in the air exceeds 3.0 ppm.

From the given information, the null and the alternative hypotheses are

Null hypothesis:

\({H_0}:\mu = 3.0\)

The mean amount of vinyl chloride in the air is 3.0 ppm

Alternative hypothesis:

\({H_a}:\mu > 3.0\)

The mean amount of vinyl chloride in the air is more significant than 3.0 ppm

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Most popular questions from this chapter

Revenue for a full-service funeral. According to the National Funeral Directors Association (NFDA), the nation's 19,000 funeral homes collected an average of \(7,180 per full-service funeral in 2014 (www.nfda.org). A random sample of 36 funeral homes reported revenue data for the current year. Among other measures, each reported its average fee for a full-service funeral. These data (in thousands of dollars) are shown in the following table.

a. What are the appropriate null and alternative hypotheses to test whether the average full-service fee of U. S. funeral homes this year is less than \)7,180?

b. Conduct the test at\(\alpha = 0.05\). Do the sample data provide sufficient evidence to conclude that the average fee this year is lower than in 2014?

c. In conducting the test, was it necessary to assume that the population of average full-service fees was normally distributed? Justify your answer

A random sample of 80 observations from a population with a population mean 198 and a population standard deviation of 15 yielded a sample mean of 190.

a. Construct a hypothesis test with the alternative hypothesis that\(\mu < 198\)at a 1% significance level. Interpret your results.

b. Construct a hypothesis test with the alternative hypothesis that

at a 1% significance level. Interpret your results.

c. State the Type I error you might make in parts a and b.

A random sample of n observations is selected from a normal population to test the null hypothesis that µ=10.Specify the rejection region for each of the following combinations of \(Ha,\alpha ,\) and n:

a.\(Ha:\)µ\( \ne 10;\alpha = .05.;n = 14\)

b.\(Ha:\)µ\( > 10;\alpha = .01;n = 24\)\(\)

c.\(Ha:\)µ\( > 10;\alpha = .10;n = 9\)

d.\(Ha:\)µ <\(10:\alpha = .01;n = 12\)

e.\(Ha:\)µ\( \ne 10;\alpha = .10;n = 20\)

f. \(Ha:\)µ<\(10;\alpha = .05;n = 4\)

Latex allergy in health care workers (cont’d). Refer to Exercise 7.120. Let \({\sigma ^2}\) represent the variance in the number of latex gloves used per week by all hospital employees. Consider testing \({H_0}:{\sigma ^2} = 100\) against\({H_a}:{\sigma ^2} \ne 100.\)

a. Give the rejection region for the test at a significance level of\(\alpha = 0.01.\)

b. Calculate the value of the test statistic.

c. Use the results, parts a and b, to make the appropriate conclusion.

A border protection avatar. The National Center for Border Security and Protection has developed the "Embodied Avatar"—a kiosk with a computer-animated border guard that uses artificial intelligence to scan passports, check fingerprints, read eye pupils, and asks questions of travellers crossing the U.S. border. (National Defense Magazine, February 2014.) Based on field tests, the avatar's developer claims that the avatar can detect deceitful speech correctly 75% of the time.

a. Identify the parameter of interest.

b. Give the null and alternative hypotheses for testing the claim made by the avatar's developer.

c. Describe a Type I error in the words of the problem.

d. Describe a Type II error in the words of the problem
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