# Chapter 6: Inferences Based on a Single Sample

11E

Tipping points in daily deal transactions? Online “daily deal” sites (e.g., Groupon) offer customers a voucher to purchase a product at discount prices. However, the number of voucher purchases must exceed a predetermined number before the deal becomes active. This key number is termed the “tipping point” in marketing. Characteristics of the tipping point were investigated in the Journal of Interactive Marketing (February 2016). A sample of 2,617 vouchers purchased from daily-deal sites in Korea had a mean tipping point of 112 sales with a standard deviation of 560 sales. The researchers want to estimate the true mean tipping point of all daily deal offerings in Korea with 95% confidence. Find and practically interpret this interval estimate.

131SE

Cell phone use by drivers. Studies have shown that driverswho use cell phones while operating a motor passenger vehicleincrease their risk of an accident. Nevertheless, driverscontinue to make cell phone calls whiledriving. A June2011 *Harris Poll*of 2,163 adults found that 60% (1,298adults) use cell phones while driving.

- Give a point estimate of
*p,*the true driver cell phone use rate (i.e., the proportion of all drivers who are usinga cell phone while operating a motor passengervehicle). - Find a 95% confidence interval for
*p*. - Give a practical interpretation of the interval, part b.
- Determine the margin of error in the interval if thenumber of adults in the survey is doubled.

132SE

Salmonella poisoning from eating an ice cream bar.Recently, a case of salmonella (bacterial) poisoning wastraced to a particular brand of ice cream bar, and themanufacturer removed the bars from the market. Despitethis response, many consumers refused to purchase *any*brand of ice cream bars for some period of time after the event (McClave, personal consulting). One manufacturerconducted a survey of consumers 6 months after theoutbreak. A sample of 244 ice cream bar consumers wascontacted, and 23 respondents indicated that they wouldnot purchase ice cream bars because of the potential forfood poisoning.

- What is the point estimate of the true fraction of the entiremarket who refuse to purchase bars 6 months after the out-break?
- Is the sample size large enough to use the normalapproximation for the sampling distribution of the estimator of the binomial probability? Justify your response.
- Construct a 95% confidence interval for the true proportionof the market who still refuses to purchase icecream bars 6 months after the event.
- Interpret both the point estimate and confidence interval in terms of this application.

136SE

Accountants’ salary survey. Each year, *ManagementAccounting*reports the results of a salary survey of themembers of the Institute of Management Accountants(IMA). One year, the 2,112 members responding had a salarydistribution with a 20th percentile of $35,100; a medianof $50,000; and an 80th percentile of $73,000.

- Use this information to determine the minimum samplesize that could be used in next year’s survey toestimate the mean salary of IMA members towithin$2,000 with 98% confidence. [
*Hint*: To estimate*s,*first applyChebyshev’s Theorem to find*k*such thatat least 60% of the data fall within*k*standard deviations of ${\mathbf{\mu}}$. Then find data-custom-editor="chemistry" ${\mathbf{s}}{\mathbf{\approx}}$(80^{th}percentile–20^{th}percentile)/*2k*.] - Explain how you estimated the standard deviation requiredfor the sample size calculation.
- List any assumptions you make.

138SE

A sampling dispute goes to court. Sampling of Medicare and Medicaid claims by the federal and state agencies who administer those programs has become common practice to determine whether providers of those services are submitting valid claims. (See the Statistics in Action for this chapter.) The reliability of inferences based on those samples depends on the methodology used to collect the sample of claims. Consider estimating the true proportion, *p*, of the population of claims that are invalid. (Invalid claims should not have been reimbursed by the agency.) Of course, to estimate a binomial parameter, *p*, within a given level of precision we use the formula provided in Section 6.5 to determine the necessary sample size. In a recent actual case, the statistician determined a sample size large enough to ensure that the bound on the error of the estimate would not exceed 0.05, using a 95% confidence interval. He did so by assuming that the true error rate was, which, as discussed in Section 6.5, provides the maximum sample size needed to achieve the desired bound on the error.

a. Determine the sample size necessary to estimate *p* to within .05 of the true value using a 95% confidence interval.

b. After the sample was selected and the sampled claims were audited, it was determined that the estimated error rate was and a 95% confidence interval for *p* was (0.15, 0.25). Was the desired bound on the error of the estimate met?

c. An economist hired by the Medicare provider noted that, since the desired bound on the error of .05 is equal to 25% of the estimated invalid claim rate, the “true” bound on the error was .25, not .05. He argued that a significantly larger sample would be necessary to meet the “relative error” (the bound on the error divided by the error rate) goal of .05, and that the statistician’s use of the “absolute error” of .05 was inappropriate, and more sampling was required. The statistician argued that the relative error was a moving target, since it depends on the sample estimate of the invalid claim rate, which cannot be known prior to selecting the sample. He noted that if the estimated invalid claim rate turned out to be larger than .5, the relative error would then be lower than the absolute error bound. As a consequence, the case went to trial over the relative vs. absolute error dispute. Give your opinion on the matter. [Note: The Court concluded that “absolute error was the fair and accurate measure of the margin of error.” As a result, a specified absolute bound on the error continues to be the accepted method for determining the sample size necessary to provide a reliable estimate of Medicare and Medicaid providers’ claim submission error rates.]

139SE

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 |

- 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.
- 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?
- 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.

13E

College dropout study. Refer to the American Economic Review (December 2008) study of college dropouts, Exercise 2.79 (p. 111). Recall that one factor thought to influence the college dropout decision was expected GPA for a student who studied 3 hours per day. In a representative sample of 307 college students who studied 3 hours per day, the mean GPA was$\overline{x}{\text{}}{=}{\text{}}{\mathbf{3}}{.}{\mathbf{11}}$ and the standard deviation was ${\mathit{s}}{\mathbf{=}}{\mathbf{0}}{\mathbf{.66}}$. Of interest is, the true mean GPA of all college students who study 3 hours per day.

a. Give a point estimate for ${\mathit{\mu}}$.

b. Give an interval estimate for ${\mathit{\mu}}$. Use a confidence coefficient of .98.

c. Comment on the validity of the following statement: “98% of the time, the true mean GPA will fall in the interval computed in part b.”

d. It is unlikely that the GPA values for college students who study 3 hours per day are normally distributed. In fact, it is likely that the GPA distribution is highly skewed. If so, what impact, if any, does this have on the validity of inferences derived from the confidence interval?

15E

Unethical corporate conduct. How complicit are entrylevel accountants in carrying out an unethical request from their superiors? This was the question of interest in a study published in the journal Behavioral Research in Accounting (July 2015). A sample of 86 accounting graduate students participated in the study. After asking the subjects to perform what is clearly an unethical task (e.g., to bribe a customer), the researchers measured each subject’s intention to comply with the unethical request score. Scores ranged from -1.5 (intention to resist the unethical request) to 2.5 (intention to comply with the unethical request). Summary statistics on the 86 scores follow: $\overline{\mathbf{x}}{\mathbf{=}}{\mathbf{2}}{\mathbf{.42}}{\mathbf{,}}{\mathit{s}}{\mathbf{=}}{\mathbf{2}}{\mathbf{.84}}$.

a. Estimate ${\mathit{\mu}}$, the mean intention to comply score for the population of all entry-level accountants, using a 90% confidence interval.

b. Give a practical interpretation of the interval, part a.

c. Refer to part a. What proportion of all similarly constructed confidence intervals (in repeated sampling) will contain the true value of ${\mathit{\mu}}$?

d. Compute the interval, $\overline{\mathbf{x}}{\mathbf{\pm}}{\mathbf{2}}{\mathit{s}}$. How does the interpretation of this interval differ from that of the confidence interval, part a?

16E

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.

26E

Let${{\mathit{t}}}_{{\mathbf{0}}}$ be a specific value of *t*. Use Table III in Appendix D to find${{\mathit{t}}}_{{\mathbf{0}}}$ values such that the following statements are true.

a. ${\mathbf{{\rm P}}}{\left(t\ge {t}_{0}\right)}{\mathbf{=}}{\mathbf{0}}{\mathbf{.025}}$where ${\mathbf{\text{df}}}{\mathbf{=}}{\mathbf{11}}$

b.${\mathbf{{\rm P}}}{\left(t\ge {t}_{0}\right)}{\mathbf{=}}{\mathbf{0}}{\mathbf{.01}}$ where${\mathbf{\text{df}}}{\mathbf{=}}{\mathbf{9}}$

c.${\mathit{{\rm P}}}{\left(t\ge {t}_{0}\right)}{\mathbf{=}}{\mathbf{0}}{\mathbf{.005}}$ where${\mathbf{\text{df}}}{\mathbf{=}}{\mathbf{6}}$

d.${\mathbf{{\rm P}}}{\left(t\ge {t}_{0}\right)}{\mathbf{=}}{\mathbf{0}}{\mathbf{.05}}$ where${\mathbf{\text{df}}}{\mathbf{=}}{\mathbf{18}}$