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Chapter 6: Q116SE (page 375)

Lead and copper in drinking water. Periodically, the Hillsborough County (Florida) Water Department tests the drinking water of homeowners for contaminants such as lead and copper. The lead and copper levels in water specimens collected for a sample of 10 residents of the Crystal Lakes Manors subdivision are shown below, followed by a Minitab printout analysing the data.

a. Locate a 90% confidence interval for the mean lead level in water specimens from Crystal Lakes Manors on the printout.

b. Locate a 90% confidence interval on the printout for the mean copper level in water specimens from Crystal Lakes Manors.

c. Interpret the intervals, parts a and b, in the words of the problem.

d. Discuss the meaning of the phrase “90% confident.”

Short Answer

Expert verified
  1. From the Minitab output, the 90% confidence interval for the mean lead in water specimen from Crystal Lakes Manors is (0.61,5.16).
  2. From the Minitab output, the 90% confidence interval for the mean copper level in water specimen from Crystal Lakes Manors is (0.2637, 0.5529).
  3. It is 90% confident that the true mean lead level in water specimens from Crystal Lakes Manors lies in between 0.61 and 5.16. It can conclude that 90% are confident that the true mean lead level in water specimens from Crystal Lakes Manors lies in between 0.2637 and 0.5529.
  4. The phrase “90% confident” indicates that 90% of all similarly constructed intervals contain the true mean.

Step by step solution

01

Given information

The lead and copper levels in water specimens collected for a sample of 10 residents of the Crystal Lakes Manors subdivision are given as follows:

Also, the Minitab output is given as:

02

(a) Finding the 90% confidence interval for the mean lead level in the water

From the Minitab output, the 90% confidence interval for the mean lead in water specimen from Crystal Lakes Manors is (0.61,5.16). It is 90% confident that the mean lead in the water specimen lies between 0.61 mg/l and 5.16 mg/l.

03

(b) Finding the 90% confidence interval for the mean copper level in the water

From the Minitab output, the 90% confidence interval for the mean copper level in water specimen from Crystal Lakes Manors is (0.2637, 0.5529). It is 90% confident that the mean copper level in the water specimen lies between 0.2637 mg/l and 0.5529 mg/l.

04

(c) Interpretation of the confidence interval for the mean lead level and mean copper level in the water

It is 90% confident that the true mean lead level in water specimens from Crystal Lakes Manors lies in between 0.61 and 5.16. It can conclude that 90% are confident that the true mean lead level in water specimens from Crystal Lakes Manors lies in between 0.2637 and 0.5529.

05

(d) Meaning of “90% confident”

The phrase “90% confident” indicates that 90% of all similarly constructed intervals contain the true mean.

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Most popular questions from this chapter

Preventing the production of defective items. It costs more toproduce defective items—because they must be scrappedor reworked—than it does to produce non-defective items.This simple fact suggests that manufacturers shouldensurethe quality of their products by perfecting theirproduction processes rather than through inspection of finishedproducts (Out of the Crisis,Deming, 1986). In orderto better understand a particular metal-stamping process, the manufacturer wishes to estimate the mean length of itemsproduced by the process during the past 24 hours.

a. How many parts should be sampled in order to estimatethe population means to within .1 millimetre (mm)with 90% confidence? Previous studies of this machinehave indicated that the standard deviation of lengthsproduced by the stamping operation is about 2 mm.

b. Time permits the use of a sample size no larger than100. If a 90% confidence interval for is constructedusing n= 100, will it be wider or narrower than wouldhave been obtained using the sample size determined in

part a? Explain.

c. If management requires that μbe estimated to within.1 mm and that a sample size of no more than 100 beused, what is (approximately) the maximum confidencelevel that could be attained for a confidence interval

Does that meet management's specifications?

Eye shadow, mascara, and nickel allergies. Pigmented makeup products like mascara and eye shadow may contain metal (e.g., nickel) allergens. Is a nickel allergy more likely to occur in women who report cosmetic dermatitis from using eye shadow or mascara? This was the question of interest in a paper published in the Journal of the European Academy of Dermatology and Venereology (June 2010). In a sample of 131 women with cosmetic dermatitis from using eye shadow, 12 were diagnosed with a nickel allergy. In a sample of 250 women with cosmetic dermatitis from using mascara, 25 were diagnosed with a nickel allergy.

a. Compute a 95% confidence interval for the proportion of women with cosmetic dermatitis from using eye shadow who have a nickel allergy. Interpret the result.

b. Compute a 95% confidence interval for the proportion of women with cosmetic dermatitis from using mascara who have a nickel allergy. Interpret the result.

c. Suppose you are informed that the true proportion with a nickel allergy for one of the two groups (eye shadow or mascara) is .12. Can you determine which group is referenced? Explain.

Employees with substance abuse problems. According to the New Jersey Governor’s Council for a Drug-Free Workplace Report, 50 of the 72 sampled businesses that are members of the council admitted that they had employees with substance abuse problems. At the time of the survey, 251 New Jersey businesses were members of the Governor’s Council. Use the finite population correction factor to find a 95% confidence interval for the proportion of all New Jersey Governor’s Council business members who have employees with substance abuse problems. Interpret the resulting interval.

Scallops, sampling, and the law. Interfaces (March–April 1995) presented the case of a ship that fishes for scallops off the coast of New England. In order to protect baby scallops from being harvested, the U.S. Fisheries and Wildlife Service requires that “the average meat per scallop weigh at least 136 of a pound.” The ship was accused of violating this weight standard. Author Arnold Barnett lays out the scenario:

The vessel arrived at a Massachusetts port with 11,000 bags of scallops, from which the harbormaster randomly selected 18 bags for weighing. From each such bag, his agents took a large scoopful of scallops; then, to estimate the bag’s average meat per scallop, they divided the total weight of meat in the scoopful by the number of scallops it contained. Based on the 18 [numbers] thus generated, the harbormaster estimated that each of the ship’s scallops possessed an average of 139 of a pound of meat (that is, they were about seven percent lighter than the minimum requirement). Viewing this outcome as conclusive evidence that the weight standard had been violated, federal authorities at once confiscated 95 percent of the catch (which they then sold at auction). The fishing voyage was thus transformed into a financial catastrophe for its participants. The actual scallop weight measurements for each of the 18 sampled bags are listed in the table below. For ease of exposition, Barnett expressed each number as a multiple of of a pound, the minimum permissible average weight per scallop. Consequently, numbers below 1 indicate individual bags that do not meet the standard. The ship’s owner filed a lawsuit against the federal government, declaring that his vessel had fully complied with the weight standard. A Boston law firm was hired to represent the owner in legal proceedings, and Barnett was retained by the firm to provide statistical litigation support and, if necessary, expert witness testimony.

0.93

0.88

0.85

0.91

0.91

0.84

0.90

0.98

0.88

0.89

0.98

0.87

0.91

0.92

0.99

1.14

1.06

0.93

  1. Recall that the harbormaster sampled only 18 of the ship’s 11,000 bags of scallops. One of the questions the lawyers asked Barnett was, “Can a reliable estimate of the mean weight of all the scallops be obtained from a sample of size 18?” Give your opinion on this issue.
  2. As stated in the article, the government’s decision rule is to confiscate a catch if the sample mean weight of the scallops is less than 136 of a pound. Do you see any flaws in this rule?
  3. Develop your own procedure for determining whether a ship is in violation of the minimum-weight restriction. Apply your rule to the data. Draw a conclusion about the ship in question.

Shopping on Black Friday. The day after Thanksgiving— called Black Friday—is one of the largest shopping days in the United States. Winthrop University researchers conducted interviews with a sample of 38 women shopping on Black Friday to gauge their shopping habits and reported the results in the International Journal of Retail and Distribution Management (Vol. 39, 2011). One question was, “How many hours do you usually spend shopping on Black Friday?” Data for the 38 shoppers are listed in the accompanying table.

a. Describe the population of interest to the researchers.

b. What is the quantitative variable of interest to the researchers?

c. Use the information in the table to estimate the population mean number of hours spent shopping on Black Friday with a 95% confidence interval.

d. Give a practical interpretation of the interval.

e. A retail store advertises that the true mean number of hours spent shopping on Black Friday is 5.5 hours. Can the store be sued for false advertising? Explain.
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