t-models, part I Using the t tables, software, or a calculator, estimate a) the critical

Salmon A specialty food company sells whole King Salmon to various customers. The mean weight of these salmon is 35 pounds with a standard deviation of 2 pounds. The company ships them to restaurants in boxes of 4 salmon, to grocery stores in cartons of 16 salmon, and to discount outlet stores in pallets of 100 salmon. To forecast costs, the shipping department needs to estimate the standard deviation of the mean weight of the salmon in each type of shipment a) Find the standard deviations of the mean weight of the salmon in each type of shipment. b) The distribution of the salmon weights turns out to be skewed to the high end. Would the distribution of shipping weights be better characterized by a Normal model for the boxes or pallets? Explain.

LSAT The LSAT (a test taken for law school admission) has a mean score of 151 with a standard deviation of 9 and a unimodal, symmetric distribution of scores. A test preparation organization teaches small classes of 9 students at a time. A larger organization teaches classes of 25 students at a time. Both organizations publish the mean scores of all their classes. a) What would you expect the distribution of mean class scores to be for each organization? b) If either organization has a graduating class with a mean score of 160, theyll take out a full-page ad in the local school paper to advertise. Which organization is more likely to have that success? Explain. c) Both organizations advertise that if any class has an average score below 145, theyll pay for everyone to retake the LSAT. Which organization is at greater risk to have to pay?

t-models, part I Using the t tables, software, or a calculator, estimate a) the critical value of t for a 90% confidence interval with df = 17. b) the critical value of t for a 98% confidence interval with df = 88. c) the P-value for t 2.09 with 4 degrees of freedom. d) the P-value for _ t _ 7 1.78 with 22 degrees of freedom.

t-models, part II Using the t tables, software, or a calculator, estimate a) the critical value of t for a 95% confidence interval with df = 7. b) the critical value of t for a 99% confidence interval with df = 102. c) the P-value for t 2.19 with 41 degrees of freedom. d) the P-value for _ t _ 7 2.33 with 12 degrees of freedom.

t-models, part III Describe how the shape, center, and spread of t-models change as the number of degrees of freedom increases.

t-models, part IV (last one!) Describe how the critical value of t for a 95% confidence interval changes as the number of degrees of freedom increases.

Cattle Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the mean weight gain this supplement produces has a margin of error of {11 pounds. Some students wrote the following conclusions. Did anyone interpret the interval correctly? Explain any misinterpretations. a) 95% of the cows studied gained between 45 and 67 pounds. b) Were 95% sure that a cow fed this supplement will gain between 45 and 67 pounds. c) Were 95% sure that the average weight gain among the cows in this study was between 45 and 67 pounds. d) The average weight gain of cows fed this supplement will be between 45 and 67 pounds 95% of the time. e) If this supplement is tested on another sample of cows, there is a 95% chance that their average weight gain will be between 45 and 67 pounds.

Teachers Software analysis of the salaries of a random sample of 288 Nevada teachers produced the confidence interval shown below. Which conclusion is correct? Whats wrong with the others? t-Interval for m: with 90.00% Confidence, 38944 6 m1TchPay2 6 42893 a) If we took many random samples of 288 Nevada teachers, about 9 out of 10 of them would produce this confidence interval. b) If we took many random samples of Nevada teachers, about 9 out of 10 of them would produce a confidence interval that contained the mean salary of all Nevada teachers. c) About 9 out of 10 Nevada teachers earn between $38,944 and $42,893. d) About 9 out of 10 of the teachers surveyed earn between $38,944 and $42,893. e) We are 90% confident that the average teacher salary in the United States is between $38,944 and $42,893.

Meal plan After surveying students at Dartmouth College, a campus organization calculated that a 95% confidence interval for the mean cost of food for one term (of three in the Dartmouth trimester calendar) is ($1102, $1290). Now the organization is trying to write its report and is considering the following interpretations. Comment on each. a) 95% of all students pay between $1102 and $1290 for food. b) 95% of the sampled students paid between $1102 and $1290. c) Were 95% sure that students in this sample averaged between $1102 and $1290 for food. d) 95% of all samples of students will have average food costs between $1102 and $1290. e) Were 95% sure that the average amount all students pay is between $1102 and $1290.

Snow Based on meteorological data for the past century, a local TV weather forecaster estimates that the regions average winter snowfall is 23_, with a margin of error of {2 inches. Assuming he used a 95% confidence interval, how should viewers interpret this news? Comment on each of these statements: a) During 95 of the last 100 winters, the region got between 21_ and 25_ of snow. b) Theres a 95% chance the region will get between 21_ and 25_ of snow this winter. c) There will be between 21_ and 25_ of snow on the ground for 95% of the winter days. d) Residents can be 95% sure that the areas average snowfall is between 21_ and 25_. e) Residents can be 95% confident that the average snowfall during the last century was between 21_ and 25_ per winter.

Pulse rates A medical researcher measured the pulse rates (beats per minute) of a sample of randomly selected adults and found the following Students t-based confidence interval: With 95.00% Confidence, 70.887604 6 m1Pulse2 6 74.497011 a) Explain carefully what the software output means. b) Whats the margin of error for this interval? c) If the researcher had calculated a 99% confidence interval, would the margin of error be larger or smaller? Explain.

Crawling Data collected by child development scientists produced this confidence interval for the average age (in weeks) at which babies begin to crawl: t-Interval for m 29.202 6 m1age2 6 31.844 (95.00% Confidence): a) Explain carefully what the software output means. b) What is the margin of error for this interval? c) If the researcher had calculated a 90% confidence interval, would the margin of error be larger or smaller? Explain.

Home sales The housing market has recovered slowly from the economic crisis of 2008. Recently, in one large community, realtors randomly sampled 36 bids from potential buyers to estimate the average loss in home value. The sample showed the average loss was $9560 with a standard deviation of $1500. a) What assumptions and conditions must be checked before finding a confidence interval? How would you check them? b) Find a 95% confidence interval for the mean loss in value per home. c) Interpret this interval and explain what 95% confidence means in this context.

Home sales again In the previous exercise, you found a 95% confidence interval to estimate the average loss in home value. a) Suppose the standard deviation of the losses had been $3000 instead of $1500. What would the larger standard deviation do to the width of the confidence interval (assuming the same level of confidence)? b) Your classmate suggests that the margin of error in the interval could be reduced if the confidence level were changed to 90% instead of 95%. Do you agree with this statement? Why or why not? c) Instead of changing the level of confidence, would it be more statistically appropriate to draw a bigger sample?

CEO compensation A sample of 20 CEOs from the Forbes 500 shows total annual compensations ranging from a minimum of $0.1 to $62.24 million. The average for these 20 CEOs is $7.946 million. Heres a histogram: Based on these data, a computer program found that a 95% confidence interval for the mean annual compensation of all Forbes 500 CEOs is (1.69, 14.20) $ million. Why should you be hesitant to trust this confidence interval?

Credit card charges A credit card company takes a random sample of 100 cardholders to see how much they charged on their card last month. Heres a histogram. A computer program found that the resulting 95% confidence interval for the mean amount spent in March 2005 is 1- +28366.84, +90691.492. Explain why the analysts didnt find the confidence interval useful, and explain what went wrong.

Normal temperature The researcher described in Exercise 11 also measured the body temperatures of that randomly selected group of adults. Here are summaries of the data he collected. We wish to estimate the average (or normal) temperature among the adult population. a) Check the conditions for creating a t-interval. b) Find a 98% confidence interval for mean body temperature. c) Explain the meaning of that interval. d) Explain what 98% confidence means in this context. e) 98.6F is commonly assumed to be normal. Do these data suggest otherwise? Explain.

Parking Hoping to lure more shoppers downtown, a city builds a new public parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period (44 weekdays), daily fees collected averaged $126, with a standard deviation of $15. a) What assumptions must you make in order to use these statistics for inference? b) Write a 90% confidence interval for the mean daily income this parking garage will generate. c) Interpret this confidence interval in context. d) Explain what 90% confidence means in this context. e) The consultant who advised the city on this project predicted that parking revenues would average $130 per day. Based on your confidence interval, do you think the consultant was correct? Why?

Normal temperatures, part II Consider again the statistics about human body temperature in Exercise 17. a) Would a 90% confidence interval be wider or narrower than the 98% confidence interval you calculated before? Explain. (Dont compute the new interval.) b) What are the advantages and disadvantages of the 98% confidence interval? c) If we conduct further research, this time using a sample of 500 adults, how would you expect the 98% confidence interval to change? Explain. d) How large a sample might allow you to estimate the mean body temperature to within 0.1 degrees with 98% confidence?

Parking II Suppose that, for budget planning purposes, the city in Exercise 18 needs a better estimate of the mean daily income from parking fees. a) Someone suggests that the city use its data to create a 95% confidence interval instead of the 90% interval first created. How would this interval be better for the city? (You need not actually create the new interval.) b) How would the 95% interval be worse for the planners? c) How could they achieve an interval estimate that would better serve their planning needs? d) How many days worth of data should they collect to have 95% confidence of estimating the true mean to within $3?

Speed of light In 1882 Michelson measured the speed of light (usually denoted c as in Einsteins famous equation E = mc2). His values are in km/sec and have 299,000 subtracted from them. He reported the results of 23 trials with a mean of 756.22 and a standard deviation of 107.12. a) Find a 95% confidence interval for the true speed of light from these statistics. b) State in words what this interval means. Keep in mind that the speed of light is a physical constant that, as far as we know, has a value that is true throughout the universe. c) What assumptions must you make in order to use your method?

Better light After his first attempt to determine the speed of light (described in Exercise 21), Michelson conducted an improved experiment. In 1897 he reported results of 100 trials with a mean of 852.4 and a standard deviation of 79.0. a) What is the standard error of the mean for these data? b) Without computing it, how would you expect a 95% confidence interval for the second experiment to differ from the confidence interval for the first? Note at least three specific reasons why they might differ, and indicate the ways in which these differences would change the interval. c) According to Stigler (who reports these values), the true speed of light is 299,710.5 km>sec, corresponding to a value of 710.5 for Michelsons 1897 measurements. What does this indicate about Michelsons two experiments? Explain, using your confidence interval.

Departures What are the chances your flight will leave on time? The U.S. Bureau of Transportation Statistics of the Department of Transportation publishes information about airline performance. Here are a histogram and summary statistics for the percentage of flights departing on time each month from 1995 thru September 2011. (www.transtats.bts.gov/HomeDrillChart.asp) There is no evidence of a trend over time. a) Check the assumptions and conditions for inference. b) Find a 90% confidence interval for the true percentage of flights that depart on time. c) Interpret this interval for a traveler planning to fly.

Arrivals Will your flight get you to your destination on time? The U.S. Bureau of Transportation Statistics reported the percentage of flights that were late each month from 1995 through September of 2011. Heres a histogram, along with some summary statistics: We can consider these data to be a representative sample of all months. There is no evidence of a time trend 1r = 0.072. a) Check the assumptions and conditions for inference about the mean. b) Find a 99% confidence interval for the true percentage of flights that arrive late. c) Interpret this interval for a traveler planning to fly.

Home prices In 2011, the average home in the region of the country studied in Exercise 13 lost $9010. Was the community studied in Exercise 13 unusual? Use a t-test to decide if the average loss observed was significantly different from the regional average.

Home prices II Suppose the standard deviation of home price losses had been $3000, as in Exercise 14? What would your conclusion be then?

For Example, and look This chapters For Examples looked at mirex contamination in farmed salmon. We first found a 95% confidence interval for the mean concentration to be 0.0834 to 0.0992 parts per million. Later we rejected the null hypothesis that the mean did not exceed the EPAs recommended safe level of 0.08 ppm based on a P-value of 0.0027. Explain how these two results are consistent. Your explanation should discuss the confidence level, the P-value, and the decision.

Hot Dogs A nutrition lab tested 40 hot dogs to see if their mean sodium content was less than the 325 mg upper limit set by regulations for reduced sodium franks. The lab failed to reject the hypothesis that the hot dogs did not meet this requirement, with a P-value of 0.142. A 90% confidence interval estimated the mean sodium content for this kind of hot dog at 317.2 to 326.8 mg. Explain how these two results are consistent. Your explanation should discuss the confidence level, the P-value, and the decision.

Pizza A researcher tests whether the mean cholesterol level among those who eat frozen pizza exceeds the value considered to indicate a health risk. She gets a P-value of 0.07. Explain in this context what the 7% represents.

Golf balls The United States Golf Association (USGA) sets performance standards for golf balls. For example, the initial velocity of the ball may not exceed 250 feet per second when measured by an apparatus approved by the USGA. Suppose a manufacturer introduces a new kind of ball and provides a sample for testing. Based on the mean speed in the test, the USGA comes up with a P-value of 0.34. Explain in this context what the 34% represents.

TV safety The manufacturer of a metal stand for home TV sets must be sure that its product will not fail under the weight of the TV. Since some larger sets weigh nearly 300 pounds, the companys safety inspectors have set a standard of ensuring that the stands can support an average of over 500 pounds. Their inspectors regularly subject a random sample of the stands to increasing weight until they fail. They test the hypothesis H0: m = 500 against HA: m 7 500, using the level of significance a = 0.01. If the sample of stands fails to pass this safety test, the inspectors will not certify the product for sale to the general public. a) Is this an upper-tail or lower-tail test? In the context of the problem, why do you think this is important? b) Explain what will happen if the inspectors commit a Type I error. c) Explain what will happen if the inspectors commit a Type II error.

Catheters During an angiogram, heart problems can be examined via a small tube (a catheter) threaded into the heart from a vein in the patients leg. Its important that the company that manufactures the catheter maintain a diameter of 2.00 mm. (The standard deviation is quite small.) Each day, quality control personnel make several measurements to test H0: m = 2.00 against HA: m _ 2.00 at a significance level of a = 0.05. If they discover a problem, they will stop the manufacturing process until it is corrected. a) Is this a one-sided or two-sided test? In the context of the problem, why do you think this is important? b) Explain in this context what happens if the quality control people commit a Type I error. c) Explain in this context what happens if the quality control people commit a Type II error.

TV safety revisited The manufacturer of the metal TV stands in Exercise 31 is thinking of revising its safety test. a) If the companys lawyers are worried about being sued for selling an unsafe product, should they increase or decrease the value of a? Explain. b) In this context, what is meant by the power of the test? c) If the company wants to increase the power of the test, what options does it have? Explain the advantages and disadvantages of each option.

Catheters again The catheter company in Exercise 32 is reviewing its testing procedure. a) Suppose the significance level is changed to a = 0.01. Will the probability of a Type II error increase, decrease, or remain the same? b) What is meant by the power of the test the company conducts? c) Suppose the manufacturing process is slipping out of proper adjustment. As the actual mean diameter of the catheters produced gets farther and farther above the desired 2.00 mm, will the power of the quality control test increase, decrease, or remain the same? d) What could they do to improve the power of the test?

Marriage In 1960, census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has increased during the past 40 years. a) Write appropriate hypotheses. b) We plan to test our hypothesis by selecting a random sample of 40 men who married for the first time last year. Do you think the necessary assumptions for inference are satisfied? Explain. c) Describe the approximate sampling distribution model for the mean age in such samples. d) The men in our sample married at an average age of 24.2 years, with a standard deviation of 5.3 years. Whats the P-value for this result? e) Explain (in context) what this P-value means. f) Whats your conclusion?

Saving gas Congress regulates corporate fuel economy and sets an annual gas mileage for cars. A company with a large fleet of cars hopes to meet the 2011 goal of 30.2 mpg or better for their fleet of cars. To see if the goal is being met, they check the gasoline usage for 50 company trips chosen at random, finding a mean of 32.12 mpg and a standard deviation of 4.83 mpg. Is this strong evidence that they have attained their fuel economy goal? a) Write appropriate hypotheses. b) Are the necessary assumptions to make inferences satisfied? c) Describe the sampling distribution model of mean fuel economy for samples like this. d) Find the P-value. e) Explain what the P-value means in this context. f) State an appropriate conclusion.

Ruffles Students investigating the packaging of potato chips purchased 6 bags of Lays Ruffles marked with a net weight of 28.3 grams. They carefully weighed the contents of each bag, recording the following weights (in grams): 29.3, 28.2, 29.1, 28.7, 28.9, 28.5. a) Do these data satisfy the assumptions for inference? Explain. b) Find the mean and standard deviation of the weights. c) Create a 95% confidence interval for the mean weight of such bags of chips. d) Explain in context what your interval means. e) Comment on the companys stated net weight of 28.3 grams.

Doritos Some students checked 6 bags of Doritos marked with a net weight of 28.3 grams. They carefully weighed the contents of each bag, recording the following weights (in grams): 29.2, 28.5, 28.7, 28.9, 29.1, 29.5. a) Do these data satisfy the assumptions for inference? Explain. b) Find the mean and standard deviation of the weights. c) Create a 95% confidence interval for the mean weight of such bags of chips. d) Explain in context what your interval means. e) Comment on the companys stated net weight of 28.3 grams.

Popcorn Yvon Hopps ran an experiment to test optimum power and time settings for microwave popcorn. His goal was to find a combination of power and time that would deliver high-quality popcorn with less than 10% of the kernels left unpopped, on average. After experimenting with several bags, he determined that power 9 at 4 minutes was the best combination. a) He concluded that this popping method achieved the 10% goal. If it really does not work that well, what kind of error did Hopps make? b) To be sure that the method was successful, he popped 8 more bags of popcorn (selected at random) at this setting. All were of high quality, with the following percentages of uncooked popcorn: 7, 13.2, 10, 6, 7.8, 2.8, 2.2, 5.2. Does this provide evidence that he met his goal of an average of no more than 10% uncooked kernels? Explain.

Ski wax Bjork Larsen was trying to decide whether to use a new racing wax for cross-country skis. He decided that the wax would be worth the price if he could average less than 55 seconds on a course he knew well, so he planned to test the wax by racing on the course 8 times. a) Suppose that he eventually decides not to buy the wax, but it really would lower his average time to below 55 seconds. What kind of error would he have made? b) His 8 race times were 56.3, 65.9, 50.5, 52.4, 46.5, 57.8, 52.2, and 43.2 seconds. Should he buy the wax? Explain.

Chips Ahoy In 1998, as an advertising campaign, the Nabisco Company announced a 1000 Chips Challenge, claiming that every 18-ounce bag of their Chips Ahoy cookies contained at least 1000 chocolate chips. Dedicated Statistics students at the Air Force Academy (no kidding) purchased some randomly selected bags of cookies, and counted the chocolate chips. Some of their data are given below. (Chance, 12, no. 1[1999]) 1219 1214 1087 1200 1419 1121 1325 1345 1244 1258 1356 1132 1191 1270 1295 1135 a) Check the assumptions and conditions for inference. Comment on any concerns you have. b) Create a 95% confidence interval for the average number of chips in bags of Chips Ahoy cookies. c) What does this evidence say about Nabiscos claim? Use your confidence interval to test an appropriate hypothesis and state your conclusion.

Yogurt Consumer Reports tested 11 brands of vanilla yogurt and found these numbers of calories per serving: 130 160 150 120 120 110 170 160 110 130 90 a) Check the assumptions and conditions for inference. b) Create a 95% confidence interval for the average calorie content of vanilla yogurt. c) A diet guide claims that you will get an average of 120 calories from a serving of vanilla yogurt. What does this evidence indicate? Use your confidence interval to test an appropriate hypothesis and state your conclusion.

Jelly A consumer advocate wants to collect a sample of jelly jars and measure the actual weight of the product in the container. He needs to collect enough data to construct a confidence interval with a margin of error of no more than 2 grams with 99% confidence. The standard deviation of these jars is usually 4 grams. What do you recommend for his sample size?

A good book An English professor is attempting to estimate the mean number of novels that the student body reads during their time in college. He is conducting an exit survey with seniors. He hopes to have a margin of error of 3 books with 95% confidence. From reading previous studies, he expects a large standard deviation and is going assume it is 10. How many students should he survey?

Maze Psychology experiments sometimes involve testing the ability of rats to navigate mazes. The mazes are classified according to difficulty, as measured by the mean length of time it takes rats to find the food at the end. One researcher needs a maze that will take rats an average of about one minute to solve. He tests one maze on several rats, collecting the data shown. a) Plot the data. Do you think the conditions for inference are satisfied? Explain. b) Test the hypothesis that the mean completion time for this maze is 60 seconds. What is your conclusion? c) Eliminate the outlier, and test the hypothesis again. What is your conclusion? d) Do you think this maze meets the one-minute average requirement? Explain.

Braking A tire manufacturer is considering a newly designed tread pattern for its all-weather tires. Tests have indicated that these tires will provide better gas mileage and longer tread life. The last remaining test is for braking effectiveness. The company hopes the tire will allow a car traveling at 60 mph to come to a complete stop within an average of 125 feet after the brakes are applied. They will adopt the new tread pattern unless there is strong evidence that the tires do not meet this objective. The distances (in feet) for 10 stops on a test track were 129, 128, 130, 132, 135, 123, 102, 125, 128, and 130. Should the company adopt the new tread pattern? Test an appropriate hypothesis and state your conclusion. Explain how you dealt with the outlier and why you made the recommendation you did.

Arrows A team of anthropologists headed by researcher Nicole Waguespack studied the difference between stonetipped and wooden-tipped arrows. Stone arrow tips are tougher, but also take longer to make. Many cultures used both types of arrow tips, including the Apache in North America and the Tiwi in Australia. The researchers set up a compound bow with 60 lbs. of force. They shot arrows of both types into a hide-covered ballistics gel. (Waguespack, Nicole, et. al., Antiquity, 2009) a) Here are the data for seven shots at the target with a wooden tip. They measured the penetration depth in mm. Find and interpret a 95% confidence interval for the penetration depth. 216 211 192 208 203 210 203 b) Here are the penetration depths (mm) for seven shots with a stone tip. Find and interpret a 95% confidence interval for the penetration depth. 240 208 213 225 232 214 240

Accuracy The researchers in the previous problem also measured the accuracy of the two types of tips. The bow was aimed at a target and the distance was measured from the center. a) Here are the data from the six wooden-tipped shots. Find and interpret a 95% confidence interval for the measure of accuracy (measured in cm). 9.3 16.7 7.1 14 1 1.2 b) Here are the data from the six stone-tipped shots. Find and interpret a 95% interval for the measure of accuracy (measured in cm). 4.9 21.1 7 1.8 5.4 8.6

Sue me! Business professor Richard Posthuma examined the number of lawsuits filed in all 50 states from 1996 to 2003. He collected data on lawsuits filed in federal court regarding employment issues. Some states had a few hundred lawsuits, while other states had thousands. Here are the summary statistics. (Business Horizons, 2012, 55) a) Find and interpret a 90% confidence interval for the mean number of employment-related lawsuits that states might expect. b) What are the shortcomings of this interval?

Sued again Dr. Posthuma (see Exercise 49) also tabulated the total amount of the lawsuits, in 1000s of dollars. Here are the statistics. a) Find and interpret a 90% confidence interval for the expected average cost of lawsuits for states. b) What are risks associated with using this confidence interval?

Driving distance How far do professional golfers drive a ball? (For non-golfers, the drive is the shot hit from a tee at the start of a hole and is typically the longest shot.) The next page shows a histogram of the average driving distances of the 186 leading professional golfers by end of November 2011 along with summary statistics (www.pgatour.com). a) Find a 95% confidence interval for the mean drive distance. b) Interpreting this interval raises some problems. Discuss. c) The data are the mean driving distance for each golfer. Is that a concern in interpreting the interval? (Hint: Review the What Can Go Wrong warnings of Chapter 8. Chapter 8?! Yes, Chapter 8.)

Wind power Should you generate electricity with your own personal wind turbine? That depends on whether you have enough wind on your site. To produce enough energy, your site should have an annual average wind speed above 8 miles per hour, according to the Wind Energy Association. One candidate site was monitored for a year, with wind speeds recorded every 6 hours. A total of 1114 readings of wind speed averaged 8.019 mph with a standard deviation of 3.813 mph. Youve been asked to make a statistical report to help the landowner decide whether to place a wind turbine at this site. a) Discuss the assumptions and conditions for using Students t inference methods with these data. Here are some plots that may help you decide whether the methods can be used: