Sample Size. In Exercises, find the sample

Statistical Literacy and Critical Thinking In Exercises 1–3, refer to the accompanying screen display that results from a sample of 40 duration times (seconds) of eruptions of the Old Faithful geyser in Yellowstone National Park. Confidence Interval Refer to the accompanying screen display. a. Express the confidence interval in the format that uses the “less than” symbol. b. Identify the best point estimate of \(\mu\) and the margin of error. ________________ Equation Transcription: Text Transcription: mu

Problem 2BSC In Exercise, refer to the accompanying screen display that results from a sample of 40 duration times (seconds) of eruptions of the Old Faithful geyser in Yellowstone National Park. Degrees of Freedom For the accompanying screen display, a simple random sample of size n= 40 was obtained from the population of duration times (seconds) of eruptions of the Old Faithful geyser. a. What is the number of degrees of freedom that should be used for finding the critical value t?/2? ________________ b. Find the critical value t?/2 corresponding to n = 40 and a 95% confidence level. ________________ c. Give a brief general description of the number of degrees of freedom.

Statistical Literacy and Critical Thinking In Exercises 1–3, refer to the accompanying screen display that results from a sample of 40 duration times (seconds) of eruptions of the Old Faithful geyser in Yellowstone National Park Interpreting a Confidence Interval The results in the screen display are based on a 95% confidence level. Write a statement that correctly interprets the confidence interval.

Cell Phone Radiation Listed below are the measured radiation emissions (in W/kg) corresponding to these cell phones: Samsung SGH­tss9, Blackberry Storm, Blackberry Curve, Motorola Moto, T­Mobile Sidekick, Sanyo Katana Eclipse, Palm Pre, Sony Ericsson, Nokia 6085, Apple iPhone 3G S, Kyocero Neo E1100. The data are from the Environmental Working Group. The media often present reports about the dangers of cell phone radiation as a cause of cancer. Construct a 90% confidence interval estimate of the population mean. What does the result suggest about the Federal Communications Commission standard that cell phone radiation must be 1.6 W/kg or less?

Lead in Medicine Listed below are the lead concentrations (in \(\mu\) g / g ) measured in different Ayurveda medicines. Ayurveda is a traditional medical system commonly used in India. The lead concentrations listed here are from medicines manufactured in the United States. The data are based on the article “Lead, Mercury, and Arsenic in US and Indian Manufactured Ayurvedic Medicines Sold via the Internet,” by Saper et al., Journal of the American Medical Association, Vol. 300, No. 8. Use the sample data to construct a 95% confidence interval estimate of the mean of the lead concentrations for the population of all such medicines. If a safety standard requires lead concentrations less than 7 \(\mu\) g / g , does it appear that the population mean is less than that level? ________________ Equation Transcription: Text Transcription: mu mu

Ages of Race Car Drivers Listed below are the ages (years) of randomly selected race car drivers (based on data reported in USA Today). Construct a 98% confidence interval estimate of the mean age of all race car drivers.

Confidence Interval with Known \(\sigma\) . In Exercises 35–38, find the confidence interval using the known value of \(\sigma\) . Construct the confidence interval for Exercise 11 assuming that \(\sigma\) is known to be 7718.8 thousand dollars. ________________ Equation Transcription: Text Transcription: sigma sigmasigma

Problem 38BB Confidence Interval with known ?. In Exercise, find the confidence interval using the known value of ?. Construct the confidence interval for Exercise assuming that ? is known to be 2.55 chocolate chips. Exercise Chocolate Chip Cookies The Chapter Problem for Chapter 3 includes the numbers of chocolate chips in a sample of 40 Chips Ahoy regular cookies. The mean is 23.95 chocolate chips and the standard deviation is 2.55 chocolate chips. Construct a 99% confidence interval estimate of the mean number of chocolate chips in all such cookies. How does the confidence interval not contradict the fact that most of the original values do not fall between the confi­dence interval limits?

Confidence Interval with Known \(\sigma\) . In Exercises 35–38, find the confidence interval using the known value of \(\sigma\) . Outlier Effect If the first value of 3.0 in Exercise 21 is changed to 300, it becomes an outlier. Does this outlier have much of an effect on the confidence interval? How should you handle outliers when they are found in sample data sets that will be used for the construction of confidence intervals? ________________ Equation Transcription: Text Transcription: sigma sigma

Using Correct Distribution. In Exercises 5–8, assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value \(\mathrm t\ \alpha\ /\ 2\) , (b) find the critical value \(\mathrm z\ \alpha\ /\ 2\) , (c) state that neither the normal distribution nor the t distribution applies. Confidence level is 99%, \(\sigma\) is not known, and the dotplot of a sample of 40 values of professional baseball player salaries is as shown below. ________________ Equation Transcription: Text Transcription: t alpha / 2 z alpha / 2 sigma

Using Correct Distribution. In Exercises 5–8, assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value \(\mathrm t\ \alpha\ /\ 2\) , (b) find the critical value \(\mathrm z\ \alpha\ /\ 2\) , (c) state that neither the normal distribution nor the t distribution applies. Confidence level is 99%, \(\sigma\) is known to be $4,385,000, and the dotplot of 40 sample values of professional baseball players is as shown in Exercise 7. ________________ Equation Transcription: Text Transcription: t alpha / 2 z alpha / 2 sigma

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Earthquake Depths Data Set 16 in Appendix B lists the depths (km) of a sample of 50 earthquakes. Those 50 depths have a mean of 9.808 km and a standard deviation of 5.013 km. Construct a 98% confidence interval estimate of the mean of all such depths. Is the confidence interval affected by the fact that the data appear to be from a population that is not normally distributed? (Hint: If using Table A­3 to find the critical value \(\mathrm t\ \alpha\ /\ 2\) , that table does not include 49 degrees of freedom, so use the closest value for 50 degrees of freedom.) ________________ Equation Transcription: Text Transcription: t alpha / 2

Problem 25BSC Sample Size. In Exercises, find the sample size required to estimate the population mean. Mean IQ of Professional Pilots The Wechsler IQ test is designed so that the mean is 100 and the standard deviation is 15 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of professional pilots. We want to be 90% confident that our sample mean is within 3 IQ points of the true mean. The mean for this population is clearly greater than 100. The standard deviation for this population is probably less than 15 because it is a group with less variation than a group randomly selected from the general population; therefore, if we use ? = 15 we are being conserva­tive by using a value that will make the sample size at least as large as necessary. Assume then that ? = 15 and determine the required sample size. Does the sample size appear to be reasonable?

Problem 26BSC Sample Size. In Exercises, find the sample size required to estimate the population mean. Mean Grade Point Average As part of a study of grade inflation, you want to estimate the mean grade point average of all current college students in the United States. All grade point averages are to be standardized for a scale between 0 and 4. How many grade point averages must be obtained so that the sample mean is within 0.2 of the population mean? As­sume that a 99% confidence level is desired. Also assume that a pilot study showed that the population standard deviation is estimated to be 0.79. Does it make sense to collect the entire sample at your college?

Problem 27B Sample Size. In Exercises, find the sample size required to estimate the population mean. Buying a Corvette Research of selling prices for used two-year-old Corvettes reveals that they have a standard deviation of $2157 (based on data from Edmunds.com). How many selling prices must you obtain in order to estimate the mean selling price of these cars? Assume that you want 98% confidence that your sample mean is within $250 of the population mean. Is it likely that you will find that many two-year-old used Corvettes in your region?

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Mean Body Temperature Data Set 3 in Appendix B includes a sample of 106 body temperatures having a mean of \(98.20\ ^\circ\ \mathrm F\) and a standard deviation of \(0.62\ ^\circ\ \mathrm F\) . Construct a 95% confidence interval estimate of the mean body temperature for the entire population. What does the result suggest about the common belief that \(98.6\ ^\circ\ \mathrm F\) is the mean body temperature? Equation Transcription: Text Transcription: 98.20 ^o F 0.62 ^o F 98.6 ^o F

Problem 14BSC Atkins Weight Loss Program In a test of weight loss programs, 40 adults used the Atkins weight loss program. After 12 months, their mean weight loss was found to be 2.1 lb, with a standard deviation of 4.8 lb. Construct a 90% confidence interval estimate of the mean weight loss for all such subjects. Does the Atkins program appear to be effective? Does it appear to be practical?

Problem 15BSC Confidence Intervals. In Exercise, construct the confidence interval. Garlic for Reducing Cholesterol In a test of the effectiveness of garlic for lowering cholesterol, 49 subjects were treated with raw garlic. Cholesterol levels were measured before and after the treatment. The changes (before minus after) in their levels of LDL cholesterol (in mg/dL) had a mean of 0.4 and a standard deviation of 21.0 (based on data from “Effect of Raw Garlic vs Commercial Garlic Supplements on Plasma Lipid Concentrations in Adults with Moderate Hypercholesterolemia,” by Gardner et al., Archives of Internal Medicine, Vol. 167). Construct a 98% confidence interval estimate of the mean net change in LDL cholesterol after the garlic treatment. What does the confidence interval suggest about the effectiveness of garlic in reducing LDL cholesterol?

Sample Size. In Exercises 25–32, find the sample size required to estimate the population mean. Sample Size Using Sample Data Refer to Data Set 1 in Appendix B and find the maximum and minimum pulse rates for males, then use those values with the range rule of thumb to estimate \(\sigma\) . How many adult males must you randomly select and test if you want to be 95% confident that the sample mean pulse rate is within 2 beats (per minute) of the true population mean \(\mu\) . The standard deviation of the 40 pulse rates of males in Data Set 1 is 10.3 beats per minute. If, instead of using the range rule of thumb, the standard deviation of the sample is used as an estimate of \(\sigma\) , is the required sample size very different? Which result is likely to be closer to the correct sample size? ________________ Equation Transcription: Text Transcription: sigma mu sigma

Sample Size. In Exercises 25–32, find the sample size required to estimate the population mean. Sample Size Using Sample Data Refer to Data Set 16 in Appendix B and find the maximum and minimum earthquake magnitudes, then use those values with the range rule of thumb to estimate \(\sigma\) . How many earthquakes must you randomly select if you want to be 95% confident that the sample mean magnitude is within 0.2 of the true population mean \(\mu\) ? The standard deviation of the 50 earthquake magnitudes in Data Set 16 is 0.587. Find the required sample size by using the standard deviation of the sample as an estimate of \(\sigma\) . Which of the two results is likely to be better? ________________ Equation Transcription: Text Transcription: sigma mu sigma

Appendix B Data Sets. In Exercises 33 and 34, use the data sets from Appendix B. Earthquake Magnitudes Use the earthquake magnitudes listed in Data Set 16 from Appendix B and construct a 99% confidence interval estimate of the mean of all such magnitudes.

Problem 16BSC Confidence Intervals. In Exercise, construct the confidence interval. Insomnia Treatment A clinical trial was conducted to test the effectiveness of the drug zopiclone for treating insomnia in older subjects. Before treatment with zopiclone,16 subjects had a mean wake time of 102.8 min. After treatment with zopiclone, the 16 subjects had a mean wake time of 98.9 min and a standard deviation of 42.3 min (based on data from “Cognitive Behavioral Therapy vs Zopiclone for Treatment of Chronic Primary Insomnia in Older Adults,” by Sivertsen et al., Journal of the American Medical Association, Vol. 295, No. 24). Assume that the 16 sample values appear to be from a normally distributed population and construct a 98% confidence interval esti­mate of the mean wake time for a population with zopiclone treatments. What does the result suggest about the mean wake time of 102.8 min before the treatment? Does zopiclone appear to be effective?

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Harry Potter Listed below are the gross amounts (in millions of dollars) earned from box office receipts for the movie Harry Potter and the Half-Blood Prince. The movie opened on a Wednesday, and the amounts are listed in order for the first 14 days of the movie's release. Use the sample values to construct a 99% confidence interval estimate of the population mean. What is the population? Identify at least one major problem with this data set.

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Years in College Listed below are the numbers of years it took for a random sample of college students to earn bachelor's degrees (based on data from the National Center for Education Statistics). Construct a 90% confidence interval estimate of the mean time required for all college students to earn bachelor's degrees. Does the confidence interval contain the value of 4 years? Is there anything about the data that would suggest that the confidence interval might not be a good result?

Appendix B Data Sets. In Exercises 33 and 34, use the data sets from Appendix B. Second­Hand Smoke Figure 7­6 from Example 5 shows a graph of three different confidence intervals. Use the cotinine levels of smokers listed in Data Set 9 from Appendix B and construct the 95% confidence interval estimate of the population mean \(\mu\). ________________ Equation Transcription: Text Transcription: mu

Confidence Interval with Known \(\sigma\) . In Exercises 35–38, find the confidence interval using the known value of \(\sigma\) Construct the confidence interval for Exercise 9 assuming that \(\sigma\) is known to be 5.013 km. ________________ Equation Transcription: Text Transcription: sigma sigma sigma

Confidence Interval with Known \(\sigma\) . In Exercises 35–38, find the confidence interval using the known value of \(\sigma\) . Construct the confidence interval for Exercise 10 assuming that \(\sigma\) is known to be 0.366ppm. ________________ Equation Transcription: Text Transcription: sigma sigma sigma

Normality Requirement What does it mean when we say that the confidence interval methods of this section are robust against departures from normality? Does the dotplot below appear to satisfy the requirement of this section? Why or why not?

Using Correct Distribution. In Exercises 5–8, assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value \(\mathrm {t\ \alpha\ /\ 2}\) , (b) find the critical value \(\mathrm {z\ \alpha\ /\ 2}\) , (c) state that neither the normal distribution nor the t distribution applies. Confidence level is 95%, \(\sigma\) is known to be $4,385,000, and the dotplot of a sample of Red Sox baseball player salaries is as shown below. ________________ Equation Transcription: Text Transcription: t alpha / 2 z alpha / 2 sigma

Using Correct Distribution. In Exercises 5–8, assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value \(\mathrm {t\ \alpha\ /\ 2}\) , (b) find the critical value \(\mathrm {z\ \alpha\ /\ 2}\) , (c) state that neither the normal distribution nor the t distribution applies. Confidence level is 90%, \(\sigma\) is not known, and the dotplot of IQ scores of 20 randomly selected statistics instructors is as shown below. ________________ Equation Transcription: Text Transcription: t alpha / 2 z alpha / 2 sigma

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Brain Volume Listed below are brain volumes ( cm 3 ) of unrelated subjects used in a study. (See Data Set 6 in Appendix B.) Use the sample data to construct a 99% confidence interval estimate of the mean of the brain volume of the population. Given that typical brain volumes are between 950 cm 3 and 1800 cm 3 , do these values appear to be typical?

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Age Discrimination? The accompanying stemplots depict ages of applicants who were unsuccessful in winning promotion and ages of applicants who were successful in winning promotion (based on data from “Debating the Use of Statistical Evidence in Allegations of Age Discrimination” by Barry and Boland, American Statistician, Vol. 58, No. 2). Assume that the samples are simple random samples and use a 95% confidence level to construct the two confidence interval estimates of the two population means. Compare the results. What do you conclude?

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Interbreeding of Cultures Changes in head sizes over time suggest interbreeding with people from other regions. Use the data depicted in the accompanying dotplots and construct 95% confidence intervals to determine whether skull breadths (mm) appear to have changed from 4000 B.C. to 150 A.D. Explain your conclusion.

Finite Population Correction Factor If a simple random sample of size n is selected without replacement from a finite population of size N, and the sample size is more than 5% of the population size ( n > 0.05 N ) , better results can be obtained by using the finite population correction factor, which involves multiplying the margin of error E by ( N ? n ) / ( N ? 1 ) . For the sample of 100 weights of M&M candies in Data Set 20 from Appendix B, we get \(\mathrm {x\ ^{-}=0.8565\ \ g}\) g and s = 0.0518 g . First construct a 95% confidence interval estimate of \(\mu\) assuming that the population is large, then construct a 95% confidence interval estimate of the mean weight of M&Ms in the full bag from which the sample was taken. The full bag has 465 M&Ms. Compare the results. ________________ Equation Transcription: Text Transcription: ( n > 0.05 N) ( N-n ) / ( N-1 ) x ^{-}=0.8565 g s=0.518 g mu

Confidence Interval for Sample of Size n = 1 Based on the article “An Effective Confidence Interval for the Mean with Samples of Size One and Two,” by Wall, Boen, and Tweedie (American Statistician, Vol. 55, No. 2), a 95% confidence interval for ? can be found (using methods not discussed in this book) for a sample of size n = 1 randomly selected from a normally distributed population, and it can be expressed as \(\mathrm {x \pm 9.68\ |\ x\ |}\) . Use this result to construct a 95% confidence interval using only the first sample value listed in Exercise 21. How does it compare to the result found in Exercise 21 when all of the sample values are used? ________________ Equation Transcription: Text Transcription: n=1 n=1 x{+/-}9.68 | x |

Confidence Intervals. In Exercises 9–24, construct the confidence interval. Mercury in Sushi Listed below are the amounts of mercury (in parts per million, or ppm) found in tuna sushi sampled at different stores in New York City. The study was sponsored by the New York Times, and the stores (in order) are D'Agostino, Eli's Manhattan, Fairway, Food Emporium, Gourmet Garage, Grace's Marketplace, and Whole Foods. The sample mean is 0.719 ppm and the standard deviation is 0.366 ppm. Construct a 90% confidence interval estimate of the mean amount of mercury in the population.

Confidence Intervals. In Exercises 9–24, construct the confidence interval. CEO Compensation Listed below are the recent annual compensation amounts (in thousands of dollars) for a random sample of chief executive officers (Mulally from Ford, Jobs from Apple, Kent from Coca­Cola, Otellini from Intel, and McNerney from Boeing). The mean of the sample is 12,898 (thousand dollars) and the standard deviation is 7719 (thousand dollars). Construct a 95% confidence interval estimate of the mean of the population of all such chief executive officers. Is there anything notable about the sample data that might have an effect on the result?

Problem 12BSC Confidence Intervals. In Exercise, construct the confidence interval. Chocolate Chip Cookies The Chapter Problem for Chapter 3 includes the numbers of chocolate chips in a sample of 40 Chips Ahoy regular cookies. The mean is 23.95 chocolate chips and the standard deviation is 2.55 chocolate chips. Construct a 99% confidence interval estimate of the mean number of chocolate chips in all such cookies. How does the confidence interval not contradict the fact that most of the original values do not fall between the confi­dence interval limits?

Problem 28BSC Sample Size. In Exercises, find the sample size required to estimate the population mean. Facebook Time You want to estimate the mean amount of time Internet users spend on Facebook each month. How many Internet users must be surveyed in order to be 95% con­fident that your sample mean is within 15 minutes of the population mean? Based on results from a prior Nielsen survey, assume that the standard deviation of the population of monthly times spent on Facebook is 210 min. What is a major obstacle to getting a good estimate of the population mean?

Problem 29BSC Sample Size. In Exercises, find the sample size required to estimate the population mean. Sample Size Using Range Rule of Thumb You want to estimate the mean SAT score of all college applicants. First use the range rule of thumb to make a rough estimate of the standard deviation of those scores. Possible SAT scores range from 600 to 2400. Use the es­timated standard deviation to determine the sample size corresponding to a 98% confidence level and a margin of error of 100 points. What isn’t quite right with this exercise?

Problem 30BSC Sample Size. In Exercises, find the sample size required to estimate the population mean. Sample Size Using Range Rule of Thumb You want to estimate the mean amount of annual tuition being paid by current full-time college students in the United States. First use the range rule of thumb to make a rough estimate of the standard deviation of the amounts spent. It is reasonable to assume that tuition amounts range from $0 to about $45,000. Then use that estimated standard deviation to determine the sample size cor­responding to 99% confidence and a $100 margin of error. Does the resulting sample size seem practical?