Solution: A study of the ability of individuals to walk in a

Three different statistics are being considered for estimating a population characteristic. The sampling distributions of the three statistics are shown in the following illustration: Which statistic would you recommend? Explain your choice.

Why is an unbiased statistic generally preferred over a biased statistic for estimating a population characteristic? Does unbiasedness alone guarantee that the estimate will be close to the true value? Explain. Under what circumstances might you choose a biased statistic over an unbiased statistic if two statistics are available for estimating a population characteristic?

Consumption of fast food is a topic of interest to researchers in the field of nutrition. The article Effects of Fast-Food Consumption on Energy Intake and Diet Quality Among Children (Pediatrics [2004]: 112118) reported that 1720 of those in a random sample of 6212 U.S. children indicated that on a typical day they ate fast food. Estimate p, the proportion of children in the U.S. who eat fast food on a typical day

Data consistent with summary quantities in the article referenced in Exercise 9.3 on total calorie consumption on a particular day are given for a sample of children who did not eat fast food on that day and for a sample of children who did eat fast food on that day. Assume that it is reasonable to regard these samples as representative of the population of children in the United States.No Fast Food 2331 1918 1009 1730 1469 2053 2143 1981 1852 1777 1765 1827 1648 1506 2669 0000 Fast Food 2523 1758 934 2328 2434 2267 2526 1195 890 1511 875 2207 1811 1250 2117 0000 a. Use the given information to estimate the mean calorie intake for children in the United States on a day when no fast food is consumed. b. Use the given information to estimate the mean calorie intake for children in the United States on a day when fast food is consumed. c. Use the given information to estimate the produce estimates of the standard deviations of calorie intake for days when no fast food is consumed and for days when fast food is consumed.

Each person in a random sample of 20 students at a particular university was asked whether he or she is registered to vote. The responses (R
registered, N
not registered) are given here: R R N R N N R R R N R R R R R N R R R N Use these data to estimate p, the true proportion of all students at the university who are registered to vote.

A study reported in Newsweek (December 23, 1991) involved a sample of 935 smokers. Each individual received a nicotine patch, which delivers nicotine to the bloodstream but at a much slower rate than cigarettes do. Dosage was decreased to 0 over a 12-week period. Suppose that 245 of the subjects were still not smoking 6 months after treatment (this figure is consistent with information given in the article). Estimate the percentage of all smokers who, when given this treatment, would refrain from smoking for at least 6 months.

The article Sensory and Mechanical Assessment of the Quality of Frankfurters (Journal of Texture Studies [1990]: 395409) reported the following salt content (percentage by weight) for 10 frankfurters: 2.26 2.11 1.64 1.17 1.64 2.36 1.70 2.10 2.19 2.40 a. Use the given data to produce a point estimate of m, the true mean salt content for frankfurters.b. Use the given data to produce a point estimate of s2 , the variance of salt content for frankfurters. c. Use the given data to produce an estimate of s, the standard deviation of salt content. Is the statistic you used to produce your estimate unbiased?

The following data on gross efficiency (ratio of work accomplished per minute to calorie expenditure per minute) for trained endurance cyclists were given in the article Cycling Efficiency Is Related to the Percentage of Type I Muscle Fibers (Medicine and Science in Sports and Exercise [1992]: 78288): 18.3 18.9 19.0 20.9 21.4 20.5 20.1 20.1 20.8 20.5 19.9 20.5 20.6 22.1 21.9 21.2 20.5 22.6 22.6 20.5 20.6 22.1 21.9 21.2 a. Assuming that the distribution of gross energy in the population of all endurance cyclists is normal, give a point estimate of m, the population mean gross efficiency. b. Making no assumptions about the shape of the population distribution, estimate the proportion of all such cyclists whose gross efficiency is at most 20.

A random sample of n
12 four-year-old red pine trees was selected, and the diameter (in inches) of each trees main stem was measured. The resulting observations are as follows: 11.3 10.7 12.4 15.2 10.1 12.1 16.2 10.5 11.4 11.0 10.7 12.0 10.1 12.1 16.2 10.5 a. Compute a point estimate of s, the population standard deviation of main stem diameter. What statistic did you use to obtain your estimate? b. Making no assumptions about the shape of the population distribution of diameters, give a point estimate for the population median diameter. What statistic did you use to obtain the estimate? c. Suppose that the population distribution of diameter is symmetric but with heavier tails than the normal distribution. Give a point estimate of the population mean diameter based on a statistic that gives some protection against the presence of outliers in the sample. What statistic did you use? d. Suppose that the diameter distribution is normal. Then the 90th percentile of the diameter distribution is m 1.28s (so 90% of all trees have diameters less than this value). Compute a point estimate for this percentile. (Hint: First compute an estimate of m in this case; then use it along with your estimate of s from Part (a).)

A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected, and the amount of gas (in therms) used during the month of January is determined for each house. The resulting observations are as follows: 103 156 118 89 125 147 122 109 138 99 a. Let mJ denote the average gas usage during January by all houses in this area. Compute a point estimate of mJ. b. Suppose that 10,000 houses in this area use natural gas for heating. Let t denote the total amount of gas used by all of these houses during January. Estimate t using the data of Part (a). What statistic did you use in computing your estimate? c. Use the data in Part (a) to estimate p, the proportion of all houses that used at least 100 therms. d. Give a point estimate of the population median usage based on the sample of Part (a). Which statistic did you use?

For each of the following choices, explain which would result in a wider large-sample confidence interval for p: a. 90% confidence level or 95% confidence level b. n
100 or n
400

The formula used to compute a large-sample con- fidence interval for p is What is the appropriate z critical value for each of the following confidence levels? a. 95% d. 80% b. 90% e. 85% c. 99%

The use of the interval requires a large sample. For each of the following combinations of n and p, indicate whether the given interval would be appropriate. a. n
50 and p
.30 b. n
50 and p
.05 c. n
15 and p
.45 d. n
100 and p
.01 e. n
100 and p
.70 f. n
40 and p
.25 g. n
60 and p
.25 h. n
80 and p
.10

Discuss how each of the following factors affects the width of the confidence interval for p: a. The confidence level b. The sample size c. The value of p

According to an AP-Ipsos poll (June 15, 2005), 42% of 1001 randomly selected adult Americans made plans in May 2005 based on a weather report that turned out to be wrong. a. Construct and interpret a 99% confidence interval for the proportion of Americans who made plans in May 2005 based on an incorrect weather report. b. Do you think it is reasonable to generalize this estimate to other months of the year? Explain.

The article Students Increasingly Turn to Credit Cards (San Luis Obispo Tribune, July 21, 2006) reported that 37% of college freshmen and 48% of college seniors carry a credit card balance from month to month. Suppose that the reported percentages were based on random samples of 1000 college freshmen and 1000 college seniors. a. Construct a 90% confidence interval for the proportion of college freshmen who carry a credit card balance from month to month. b. Construct a 90% confidence interval for the proportion of college seniors who carry a credit card balance from month to month. c. Explain why the two 90% confidence intervals from Parts (a) and (b) are not the same width.

The article CSI Effect Has Juries Wanting More Evidence (USA Today, August 5, 2004) examines how the popularity of crime-scene investigation television shows is influencing jurors expectations of what evidence should be produced at a trial. In a survey of 500 potential jurors, one study found that 350 were regular watchers of at least one crime-scene forensics television series. a. Assuming that it is reasonable to regard this sample of 500 potential jurors as representative of potential jurors in the United States, use the given information to construct and interpret a 95% confidence interval for the true proportion of potential jurors who regularly watch at least one crime-scene investigation series. b. Would a 99% confidence interval be wider or narrower than the 95% confidence interval from Part (a)?

In a survey of 1000 randomly selected adults in the United States, participants were asked what their most favorite and what their least favorite subject was when they were in school (Associated Press, August 17, 2005). In what might seem like a contradiction, math was chosen more often than any other subject in both categories! Math was chosen by 230 of the 1000 as the favorite subject, and it was also chosen by 370 of the 1000 as the least favorite subject. a. Construct a 95% confidence interval for the proportion of U.S. adults for whom math was the favorite subject in school. b. Construct a 95% confidence interval for the proportion of U.S. adults for whom math was the least favorite subject

The report 2005 Electronic Monitoring & Surveillance Survey: Many Companies Monitoring, Recording, Videotapingand FiringEmployees (American Management Association, 2005) summarized the results of a survey of 526 U.S. businesses. The report stated that 137 of the 526 businesses had fired workers for misuse of the Internet and 131 had fired workers for email misuse. For purposes of this exercise, assume that it is reasonable to regard this sample as representative of businesses in the United States. a. Construct and interpret a 95% confidence interval for the proportion of U.S. businesses that have fired workers for misuse of the Internet. b. What are two reasons why a 90% confidence interval for the proportion of U.S. businesses that have fired workers for misuse of email would be narrower than the 95% confidence interval computed in Part (a).

In an AP-AOL sports poll (Associated Press, December 18, 2005), 394 of 1000 randomly selected U.S. adults indicated that they considered themselves to be baseball fans. Of the 394 baseball fans, 272 stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. a. Construct a 95% confidence interval for the proportion of U.S. adults that consider themselves to be baseball fans. b. Construct a 95% confidence interval for the proportion of those who consider themselves to be baseball fans that think the designated hitter rule should be expanded to both leagues or eliminated. c. Explain why the confidence intervals of Parts (a) and (b) are not the same width even though they both have a confidence level of 95%.

The article Viewers Speak Out Against Reality TV (Associated Press, September 12, 2005) included the following statement: Few people believe theres much reality in reality TV: a total of 82 percent said the shows are either totally made up or mostly distorted. This statement was based on a survey of 1002 randomly selected adults. Compute and interpret a bound on the error of estimation for the reported percentage.

One thousand randomly selected adult Americans participated in a survey conducted by the Associated Press (June, 2006). When asked Do you think it is sometimes justified to lie or do you think lying is never justified? 52% responded that lying was never justified. When asked about lying to avoid hurting someones feelings, 650 responded that this was often or sometimes OK. a. Construct a 90% confidence interval for the proportion of adult Americans who think lying is never justified. b. Construct a 90% confidence interval for the proportion of adult American who think that it is often or sometimes OK to lie to avoid hurting someones feelings. c. Based on the confidence intervals from Parts (a) and (b), comment on the apparent inconsistency in the responses given by the individuals in this sample.

The article Doctors Cite Burnout in Mistakes (San Luis Obispo Tribune, March 5, 2002) reported that many doctors who are completing their residency have financial struggles that could interfere with training. In a sample of 115 residents, 38 reported that they worked moonlighting jobs and 22 reported a credit card debt of more than $3000. Suppose that it is reasonable to consider this sample of 115 as a random sample of all medical residents in the United States. a. Construct and interpret a 95% confidence interval for the proportion of U.S. medical residents who work moonlighting jobs. b. Construct and interpret a 90% confidence interval for the proportion of U.S. medical residents who have a credit card debt of more than $3000. c. Give two reasons why the confidence interval in Part (a) is wider than the confidence interval in Part (b).

The National Geographic Society conducted a study that included 3000 respondents, age 18 to 24, in nine different countries (San Luis Obispo Tribune, November 21, 2002). The society found that 10% of the participants could not identify their own country on a blank world map. a. Construct a 90% confidence interval for the proportion who can identify their own country on a blank world map. b. What assumptions are necessary for the confidence interval in Part (a) to be valid? c. To what population would it be reasonable to generalize the confidence interval estimate from Part (a)?

Tongue Piercing May Speed Tooth Loss, Researchers Say is the headline of an article that appeared in the San Luis Obispo Tribune (June 5, 2002). The article describes a study of 52 young adults with pierced tongues. The researchers found receding gums, which can lead to tooth loss, in 18 of the participants. Construct a 95% con- fidence interval for the proportion of young adults with pierced tongues who have receding gums. What assumptions must be made for use of the z confidence interval to be appropriate?

USA Today (October 14, 2002) reported that 36% of adult drivers admit that they often or sometimes talk on a cell phone when driving. This estimate was based on data from a sample of 1004 adult drivers, and a bound on the error of estimation of 3.1% was reported. Explain how the given bound on the error can be justified.

The Gallup Organization conducts an annual survey on crime. It was reported that 25% of all households experienced some sort of crime during the past year. This estimate was based on a sample of 1002 randomly selected adults. The report states, One can say with 95% con- fidence that the margin of sampling error is  3 percentage points. Explain how this statement can be justified.

In a study of 1710 schoolchildren in Australia (Herald Sun, October 27, 1994), 1060 children indicated that they normally watch TV before school in the morning. (Interestingly, only 35% of the parents said their children watched TV before school!) Construct a 95% confidence interval for the true proportion of Australian children who say they watch TV before school. What assumption about the sample must be true for the method used to construct the interval to be valid?

A consumer group is interested in estimating the proportion of packages of ground beef sold at a particular store that have an actual fat content exceeding the fat content stated on the label. How many packages of ground beef should be tested to estimate this proportion to within .05 with 95% confidence?

Given a variable that has a t distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? a. 10 df, between 1.81 and 1.81 b. 10 df, between 2.23 and 2.23 c. 24 df, between 2.06 and 2.06 d. 24 df, between 2.80 and 2.80 e. 24 df, outside the interval from 2.80 to 2.80 f. 24 df, to the right of 2.80 g. 10 df, to the left of 1.81

The formula used to compute a confidence interval for the mean of a normal population when n is small is What is the appropriate t critical value for each of the following confidence levels and sample sizes? a. 95% confidence, n
17 b. 90% confidence, n
12 c. 99% confidence, n
24 d. 90% confidence, n
25 e. 90% confidence, n
13 f. 95% confidence, n
10

The two intervals (114.4, 115.6) and (114.1, 115.9) are confidence intervals for m
true average resonance frequency (in hertz) for all tennis rackets of a certain type. a. What is the value of the sample mean resonance frequency? b. The confidence level for one of these intervals is 90% and for the other it is 99%. Which is which, and how can you tell?

Samples of two different types of automobiles were selected, and the actual speed for each car was determined when the speedometer registered 50 mph. The resulting 95% confidence intervals for true average actual speed were (51.3, 52.7) and (49.4, 50.6). Assuming that the two sample standard deviations are identical, which confidence interval is based on the larger sample size? Explain your reasoning.

Suppose that a random sample of 50 bottles of a particular brand of cough medicine is selected and the alcohol content of each bottle is determined. Let m denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the sample of 50 results in a 95% confidence interval for m of (7.8, 9.4). a. Would a 90% confidence interval have been narrower or wider than the given interval? Explain your answer. b. Consider the following statement: There is a 95% chance that m is between 7.8 and 9.4. Is this statement correct? Why or why not? c. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% confidence interval is repeated 100 times, 95 of the resulting intervals will include m. Is this statement correct? Why or why not?

Acrylic bone cement is sometimes used in hip and knee replacements to fix an artificial joint in place. The force required to break an acrylic bone cement bond was measured for six specimens under specified conditions, and the resulting mean and standard deviation were 306.09 Newtons and 41.97 Newtons, respectively. Assuming that it is reasonable to assume that breaking force under these conditions has a distribution that is approximately normal, estimate the true average breaking force for acrylic bone cement under the specified conditions.

The article The Association Between Television Viewing and Irregular Sleep Schedules Among Children Less Than 3 Years of Age (Pediatrics [2005]: 851856) reported the accompanying 95% confidence intervals for average TV viewing time (in hours per day) for three different age groups. Age Group 95% Confidence Interval Less than 12 months (0.8, 1.0) 12 to 23 months (1.4, 1.8) 24 to 35 months (2.1, 2.5) a. Suppose that the sample sizes for each of the three age group samples were equal. Based on the given confidence intervals, which of the age group samples had the greatest variability in TV viewing time? Explain your choice. b. Now suppose that the sample standard deviations for the three age group samples were equal, but that the three sample sizes might have been different. Which of the three age group samples had the largest sample size? Explain your choice. c. The interval (.768, 1.302) is either a 90% confidence interval or a 99% confidence interval for the mean TV viewing time for children less than 12 months old. Is the confidence level for this interval 90% or 99%? Explain your choice.

Five hundred randomly selected working adults living in Calgary, Canada were asked how long, in minutes, their typical daily commute was (Calgary Herald Traffic Study, Ipsos, September 17, 2005). The resulting sample mean and standard deviation of commute time were 28.5 minutes and 24.2 minutes, respectively. Construct and interpret a 90% confidence interval for the mean commute time of working adult Calgary residents.

The article Most Canadians Plan to Buy Treats, Many Will Buy Pumpkins, Decorations and/or Costumes (Ipsos-Reid, October 24, 2005) summarized results from a survey of 1000 randomly selected Canadian residents. Each individual in the sample was asked how much he or she anticipated spending on Halloween during 2005. The resulting sample mean and standard deviation were $46.65 and $83.70 respectively. a. Explain how it could be possible for the standard deviation of the anticipated Halloween expense to be larger than the mean anticipated expense. b. Is it reasonable to think that the distribution of the variable anticipated Halloween expense is approximately normal? Explain why or why not. c. Is it appropriate to use the t confidence interval to estimate the mean anticipated Halloween expense for Canadian residents? Explain why or why not. d. If appropriate, construct and interpret a 99% confidence interval for the mean anticipated Halloween expense for Canadian residents.

Because of safety considerations, in May 2003 the Federal Aviation Administration (FAA) changed its guidelines for how small commuter airlines must estimate passenger weights. Under the old rule, airlines used 180 lb as a typical passenger weight (including carry-on luggage) in warm months and 185 lb as a typical weight in cold months. The Alaska Journal of Commerce (May 25, 2003) reported that Frontier Airlines conducted a study to estimate average passenger plus carry-on weights. They found an average summer weight of 183 lb and a winter average of 190 lb. Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 20 lb for the summer weights and 23 lb for the winter weights. a. Construct and interpret a 95% confidence interval for the mean summer weight (including carry-on luggage) of Frontier Airlines passengers. b. Construct and interpret a 95% confidence interval for the mean winter weight (including carry-on luggage) of Frontier Airlines passengers. c. The new FAA recommendations are 190 lb for summer and 195 lb for winter. Comment on these recommendations in light of the confidence interval estimates from Parts (a) and (b).

Heinz Plays Catch-up After Under-Filling Ketchup Containers is the headline of an article that appeared on CNN.com (November 30, 2000). The article stated that Heinz had agreed to put an extra 1% of ketchup into each ketchup container sold in California for a 1-year period. Suppose that you want to make sure that Heinz is in fact fulfilling its end of the agreement. You plan to take a sample of 20-oz bottles shipped to California, measure the amount of ketchup in each bottle, and then use the resulting data to estimate the mean amount of ketchup in each bottle. A small pilot study showed that the amount of ketchup in 20-oz bottles varied from 19.9 to 20.3 oz. How many bottles should be included in the sample if you want to estimate the true mean amount of ketchup to within 0.1 oz with 95% confidence?

Example 9.3 gave the following airborne times for United Airlines flight 448 from Albuquerque to Denver on 10 randomly selected days: 57 54 55 51 56 48 52 51 59 59 a. Compute and interpret a 90% confidence interval for the mean airborne time for flight 448. b. Give an interpretation of the 90% confidence level associated with the interval estimate in Part (a). c. Based on your interval in Part (a), if flight 448 is scheduled to depart at 10 A.M., what would you recommend for the published arrival time? Explain.

The authors of the paper Short-Term Health and Economic Benefits of Smoking Cessation: Low Birth Weight (Pediatrics [1999]: 13121320) investigated the medical cost associated with babies born to mothers who smoke. The paper included estimates of mean medical cost for low-birth-weight babies for different ethnic groups. For a sample of 654 Hispanic low-birth-weight babies, the mean medical cost was $55,007 and the standard error (s/ ) was $3011. For a sample of 13 Native American low-birth-weight babies, the mean and standard error were $73,418 and $29,577, respectively. Explain why the two standard errors are so different.

A study of the ability of individuals to walk in a straight line (Can We Really Walk Straight? American Journal of Physical Anthropology [1992]: 1927) reported the following data on cadence (strides per second) for a sample of n
20 randomly selected healthy men: 0.95 0.85 0.92 0.95 0.93 0.86 1.00 0.92 0.85 0.81 0.78 0.93 0.93 1.05 0.93 1.06 1.06 0.96 0.81 0.96 Construct and interpret a 99% confidence interval for the population mean cadence.

Fat contents (in percentage) for 10 randomly selected hot dogs were given in the article Sensory and Mechanical Assessment of the Quality of Frankfurters (Journal of Texture Studies [1990]: 395409). Use the following data to construct a 90% confidence interval for the true mean fat percentage of hot dogs: 25.2 21.3 22.8 17.0 29.8 21.0 25.5 16.0 20.9 19.5

Five students visiting the student health center for a free dental examination during National Dental Hygiene Month were asked how many months had passed since their last visit to a dentist. Their responses were as follows: 6 17 11 22 29 Assuming that these five students can be considered a random sample of all students participating in the free checkup program, construct a 95% confidence interval for the mean number of months elapsed since the last visit to a dentist for the population of students participating in the program.

The article First Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students (Journal of College Student Development [1999]: 599610) reported that the sample mean and standard deviation for high school grade point average (GPA) for students enrolled at a large research university were 3.73 and 0.45, respectively. Suppose that the mean and standard deviation were based on a random sample of 900 students at the university. a. Construct a 95% confidence interval for the mean high school GPA for students at this university. b. Suppose that you wanted to make a statement about the range of GPAs for students at this university. Is it reasonable to say that 95% of the students at the university have GPAs in the interval you computed in Part (a)? Explain

The following data are the calories per half-cup serving for 16 popular chocolate ice cream brands reviewed by Consumer Reports (July 1999): 270 150 170 140 160 160 160 290 190 190 160 170 150 110 180 170 Is it reasonable to use the t confidence interval to compute a confidence interval for m, the true mean calories per half-cup serving of chocolate ice cream? Explain why or why not.

The Bureau of Alcohol, Tobacco, and Firearms (BATF) has been concerned about lead levels in California wines. In a previous testing of wine specimens, lead levels ranging from 50 to 700 parts per billion were recorded (San Luis Obispo Telegram Tribune, June 11, 1991). How many wine specimens should be tested if the BATF wishes to estimate the true mean lead level for California wines to within 10 parts per billion with 95% confidence?

The article National Geographic, the Doomsday Machine, which appeared in the March 1976 issue of the Journal of Irreproducible Results (yes, there really is a journal by that nameits a spoof of technical journals!) predicted dire consequences resulting from a nationwide buildup of National Geographic magazines. The authors predictions are based on the observation that the number of subscriptions for National Geographic is on the rise and that no one ever throws away a copy of National Geographic. A key to the analysis presented in the article is the weight of an issue of the magazine. Suppose that you were assigned the task of estimating the average weight of an issue of National Geographic. How many issues should you sample to estimate the average weight to within 0.1 oz with 95% confidence? Assume that s is known to be 1 oz.

The formula described in this section for determining sample size corresponds to a confidence level of 95%. What would be the appropriate formula for determining sample size when the desired confidence level is 90%? 98%?

Click on the Sample button several more times, and notice how the confidence interval estimate changes from sample to sample. Also notice that at the bottom of the left-hand side of the display, the applet is keeping track of the proportion of all the intervals calculated so far that include the actual value of m. If we were to construct a large number of intervals, this proportion should closely approximate the capture rate for the confidence interval method. To look at more than 1 interval at a time, change the Intervals box from 1 to 100, and then click the sample button. You should see a screen similar to the one at the top of page 515, with 100 intervals is the display on the right-hand side. Again, intervals containing 100 (the value of m in this case) will be green and those that do not contain 100 will be red. Also note that the capture proportion on the left-hand side has also been updated to reflect what happened with the 100 newly generated intervals. Experiment with three other confidence levels of your choice, and then answer the following question: b. In general, is the proportion of computed t confidence intervals that contain m
100 close to the stated confi- dence level?

When the population is normal but s is unknown, we construct a confidence interval for a population mean using a t critical value rather than a z critical value. How important is this distinction? Lets investigate. Use the drop-down menu to change the box just below the method box thats says Means from t to z with s. The applet will now construct intervals using the sample standard deviation, but will use a z critical value rather than the t critical value. Use the applet to construct at least 1000 95% intervals, and then answer the following question: c. Comment on how the proportion of the computed intervals that include the actual value of the population mean compares to the stated confidence level of 95%. Is this surprising? Explain why or why not. Now experiment with some different samples sizes. What happens when n
20? n
50? n
100? Use what you have learned to write a paragraph explaining what these simulations tell you about the advisability of using a z critical value in the construction of a confidence interval for m when s is unknown.

How does the proportion of intervals constructed that include p
.3, the population proportion, compare to 95%? Does this surprise you? Explain. Now use the drop-down menu to change Large Sample z to Modified. Now the applet will construct the alternative confidence interval that is based on Pmod. Use the applet to construct a large number (at least 1000) of 95% confidence intervals.

How does the proportion of intervals constructed that include p
.3, the population proportion, compare to 95%? Is this proportion closer to 95% than was the case for the large-sample z interval?

Experiment with different combinations of values of sample size and population proportion p. Can you find a combination for which the large sample z interval has a capture rate that is close to 95%? Can you find a combination for which it has a capture rate that is even farther from 95% than it was for n
40 and p
.3? How does the modified interval perform in each of these cases?

Use the data from the random sample of 567 San Luis Obispo County signatures to construct a 95% confidence interval for the proportion of petition signatures that are valid.

How do you think that the reported figure of 16,000 verified signature for San Luis Obispo County was obtained?

Based on your confidence interval from Step 1, explain why you think that the reported figure of 16,000 verified signatures is or is not reasonable.

Write a meaningful paragraph that includes the following six terms: sample, population, confidence level, estimate, mean, margin of error. A meaningful paragraph is a coherent piece writing in an appropriate context that uses all of the listed words. The paragraph should show that you understand the meaning of the terms and their relationship to one another. A sequence of sentences that just define the terms is not a meaningful paragraph. When choosing a context, think carefully about the terms you need to use. Choosing a good context will make writing a meaningful paragraph easier.

Despite protests from civil libertarians and gay rights activists, many people favor mandatory AIDS testing of certain at-risk groups, and some people even believe that all citizens should be tested. What proportion of the adults in the United States favor mandatory testing for all citizens? To assess public opinion on this issue, researchers conducted a survey of 1014 randomly selected adult U.S. citizens (Large Majorities Continue to Back AIDS Testing, Gallup Poll Monthly [1991]: 2528). The article reported that 466 of the 1014 people surveyed believed that all citizens should be tested. Use this information to estimate p, the true proportion of all U.S. adults who favor AIDS testing of all citizens.

The article Consumers Show Increased Liking for Diesel Autos (USA Today, January 29, 2003) reported that 27% of U.S. consumers would opt for a diesel car if it ran as cleanly and performed as well as a car with a gas engine. Suppose that you suspect that the proportion might be different in your area and that you want to conduct a survey to estimate this proportion for the adult residents of your city. What is the required sample size if you want to estimate this proportion to within .05 with 95% confi- dence? Compute the required sample size first using .27 as a preliminary estimate of p and then using the conservative value of .5. How do the two sample sizes compare? What sample size would you recommend for this study?

In the article Fluoridation Brushed Off by Utah (Associated Press, August 24, 1998), it was reported that a small but vocal minority in Utah has been successful in keeping fluoride out of Utah water supplies despite evidence that fluoridation reduces tooth decay and despite the fact that a clear majority of Utah residents favor fluoridation. To support this statement, the article included the result of a survey of Utah residents that found 65% to be in favor of fluoridation. Suppose that this result was based on a random sample of 150 Utah residents. Construct and interpret a 90% confidence interval for p, the true proportion of Utah residents who favor fluoridation. Is this interval consistent with the statement that fluoridation is favored by a clear majority of residents?

Seventy-seven students at the University of Virginia were asked to keep a diary of a conversation with their mothers, recording any lies they told during these conversations (San Luis Obispo Telegram-Tribune, August 16, 1995). It was reported that the mean number of lies per conversation was 0.5. Suppose that the standard deviation (which was not reported) was 0.4. a. Suppose that this group of 77 is a random sample from the population of students at this university. Construct a 95% confidence interval for the mean number of lies per conversation for this population. b. The interval in Part (a) does not include 0. Does this imply that all students lie to their mothers? Explain.

The article Selected Characteristics of High-Risk Students and Their Enrollment Persistence (Journal of College Student Development [1994]: 5460) examined factors that affect whether students stay in college. The following summary statistics are based on data from a sample of 44 students who did not return to college after the first quarter (the nonpersisters) and a sample of 257 students who did return (the persisters):a. Consider the 44 nonpersisters as a random sample from the population of all nonpersisters at the university where the data were collected. Compute a 98% confidence interval for the mean number of hours worked per week for nonpersisters. b. Consider the 257 persisters as a random sample from the population of all persisters at the university where the data were collected. Compute a 98% confidence interval for the mean number of hours worked per week for persisters.c. The 98% confidence interval for persisters is narrower than the corresponding interval for nonpersisters, even though the standard deviation for persisters is larger than that for nonpersisters. Explain why this happened. d. Based on the interval in Part (a), do you think that the mean number of hours worked per week for nonpersisters is greater than 20? Explain.

An Associated Press article on potential violent behavior reported the results of a survey of 750 workers who were employed full time (San Luis Obispo Tribune, September 7, 1999). Of those surveyed, 125 indicated that they were so angered by a coworker during the past year that they felt like hitting the coworker (but didnt). Assuming that it is reasonable to regard this sample of 750 as a random sample from the population of full-time workers, use this information to construct and interpret a 90% con- fidence interval estimate of p, the true proportion of fulltime workers so angered in the last year that they wanted to hit a colleague.

The 1991 publication of the book Final Exit, which includes chapters on doctor-assisted suicide, caused a great deal of controversy in the medical community. The Society for the Right to Die and the American Medical Association quoted very different figures regarding the proportion of primary-care physicians who have participated in some form of doctor-assisted suicide for terminally ill patients (USA Today, July 1991). Suppose that a survey of physicians is to be designed to estimate this proportion to within .05 with 95% confidence. How many primary-care physicians should be included in a random sample?

Retailers report that the use of cents-off coupons is increasing. The Scripps Howard News Service (July 9, 1991) reported the proportion of all households that use coupons as .77. Suppose that this estimate was based on a random sample of 800 households (i.e., n
800 and p
.77). Construct a 95% confidence interval for p, the true proportion of all households that use coupons.

A manufacturer of small appliances purchases plastic handles for coffeepots from an outside vendor. If a handle is cracked, it is considered defective and must be discarded. A large shipment of plastic handles is received.The proportion of defective handles p is of interest. How many handles from the shipment should be inspected to estimate p to within 0.1 with 95% confidence?

An article in the Chicago Tribune (August 29, 1999) reported that in a poll of residents of the Chicago suburbs, 43% felt that their financial situation had improved during the past year. The following statement is from the article: The findings of this Tribune poll are based on interviews with 930 randomly selected suburban residents. The sample included suburban Cook County plus DuPage, Kane, Lake, McHenry, and Will Counties. In a sample of this size, one can say with 95% certainty that results will differ by no more than 3 percent from results obtained if all residents had been included in the poll. Comment on this statement. Give a statistical argument to justify the claim that the estimate of 43% is within 3% of the true proportion of residents who feel that their financial situation has improved.

The McClatchy News Service (San Luis Obispo Telegram-Tribune, June 13, 1991) reported on a study of violence on television during primetime hours. The following table summarizes the information reported for four networks: Mean Number of Network Violent Acts per Hour ABC 15.6 CBS 11.9 FOX 11.7 NBC 11.0 Suppose that each of these sample means was computed on the basis of viewing n
50 randomly selected primetime hours and that the population standard deviation for each of the four networks is known to be s
5. a. Compute a 95% confidence interval for the true mean number of violent acts per prime-time hour for ABC. b. Compute 95% confidence intervals for the mean number of violent acts per prime-time hour for each of the other three networks. c. The National Coalition on Television Violence claims that shows on ABC are more violent than those on the other networks. Based on the confidence intervals from Parts (a) and (b), do you agree with this conclusion? Explain.

The Chronicle of Higher Education (January 13, 1993) reported that 72.1% of those responding to a national survey of college freshmen were attending the college of their first choice. Suppose that n
500 students responded to the survey (the actual sample size was much larger). a. Using the sample size n
500, calculate a 99% con- fidence interval for the proportion of college students who are attending their first choice of college. b. Compute and interpret a 95% confidence interval for the proportion of students who are not attending their first choice of college. c. The actual sample size for this survey was much larger than 500. Would a confidence interval based on the actual sample size have been narrower or wider than the one computed in Part (a)?

Increases in worker injuries and disability claims have prompted renewed interest in workplace design and regulation. As one particular aspect of this, employees required to do regular lifting should not have to handle unsafe loads. The article Anthropometric, Muscle Strength, and Spinal Mobility Characteristics as Predictors of the Rating of Acceptable Loads in Parcel Sorting (Ergonomics [1992]: 10331044) reported on a study involving a random sample of n
18 male postal workers. The sample mean rating of acceptable load attained with a work-simulating test was found to be
9.7 kg. and the sample standard deviation was s
4.3 kg. Suppose that in the population of all male postal workers, the distribution of rating of acceptable load can be modeled approximately using a normal distribution with mean value m. Construct and interpret a 95% confidence interval for m.

The Gallup Organization conducted a telephone survey on attitudes toward AIDS (Gallup Monthly, 1991). A total of 1014 individuals were contacted. Each individual was asked whether they agreed with the following statement: Landlords should have the right to evict a tenant from an apartment because that person has AIDS. One hundred one individuals in the sample agreed with this statement. Use these data to construct a 90% confidence interval for the proportion who are in agreement with this statement. Give an interpretation of your interval.

A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate with 95% confidence to within 0.1 lb, the average force required to break the binding? Assume that s is known to be 0.8 lb.

Recent high-profile legal cases have many people reevaluating the jury system. Many believe that juries in criminal trials should be able to convict on less than a unanimous vote. To assess support for this idea, investigators asked each individual in a random sample of Californians whether they favored allowing conviction by a 102 verdict in criminal cases not involving the death penalty. The Associated Press (San Luis Obispo TelegramTribune, September 13, 1995) reported that 71% supported the 102 verdict. Suppose that the sample size for this survey was n
900. Compute and interpret a 99% confi- dence interval for the proportion of Californians who favor the 102 verdict.

The Center for Urban Transportation Research released a report stating that the average commuting distance in the United States is 10.9 mi (USA Today, August 13, 1991). Suppose that this average is actually the mean of a random sample of 300 commuters and that the sample standard deviation is 6.2 mi. Estimate the true mean commuting distance using a 99% confidence interval.

In 1991, California imposed a snack tax (a sales tax on snack food) in an attempt to help balance the state budget. A proposed alternative tax was a 12-per-pack increase in the cigarette tax. In a poll of 602 randomly selected California registered voters, 445 responded that they would have preferred the cigarette tax increase to the snack tax (Reno Gazette-Journal, August 26, 1991). Estimate the true proportion of California registered voters who preferred the cigarette tax increase; use a 95% con- fidence interval.

The confidence intervals presented in this chapter give both lower and upper bounds on plausible values for the population characteristic being estimated. In some instances, only an upper bound or only a lower bound is appropriate. Using the same reasoning that gave the large sample interval in Section 9.3, we can say that when n is large, 99% of all samples have (because the area under the z curve to the left of 2.33 is .99). Thus, is a 99% upper confidence bound for m. Use the data of Example 9.8 to calculate the 99% upper confidence bound for the true wait time for bypass patients in Ontario.

The Associated Press (December 16, 1991) reported that in a random sample of 507 people, only 142 correctly described the Bill of Rights as the first 10 amendments to the U.S. Constitution. Calculate a 95% confidence interval for the proportion of the entire population that could give a correct description.

When n is large, the statistic s is approximately unbiased for estimating s and has approximately a normal distribution. The standard deviation of this statistic when the population distribution is normal is which can be estimated by . A large-sample confidence interval for the population standard deviation s is then s  1z critical value2 s 12n s 12n ss
s 12n x 2.33 s 1n m x 2.33 s 1n Use the data of Exercise 9.67 to obtain a 95% confidence interval for the true standard deviation of commuting distance. 9

The interval from 2.33 to 1.75 captures an area of .95 under the z curve. This implies that another largesample 95% confidence interval for m has lower limit and upper limit . Would you recommend using this 95% interval over the 95% interval discussed in the text? Explain. (Hint: Look at the width of each interval.)

The eating habits of 12 bats were examined in the article Foraging Behavior of the Indian False Vampire Bat (Biotropica [1991]: 6367). These bats consume insects and frogs. For these 12 bats, the mean time to consume a frog was min. Suppose that the standard deviation was s
7.7 min. Construct and interpret a 90% confidence interval for the mean suppertime of a vampire bat whose meal consists of a frog. What assumptions must be reasonable for the one-sample t interval to be appropriate?