Celebriries and the Law Here is a 95% confidence interval estimate of the proportion ofadults who say that the law goes easy on celebrities: 0.692 <p < 0.748 (based on data froma Rasmussen Reports survey). What is the best point estimate of the proportion of adults in thepopulation who say that the law goes easy on celebrities?

Online News In a Harris poll of 2036 adults, 40% said that they prefer to get their news online.Construct a 95% confidence interval estimate of the percentage of all adults who say that they prefer toget their news online. Can we safely say that fewer than 50% of adults prefer to get their news online?

Flight Arrivals. Listed below are the arrival delay times (min) of randomly selected American Airlines flights that departed from JFK in New York bound for LAX in Los Angeles. Negative values correspond to flights that arrived early and ahead of the scheduled arrival time. Use these values for Exercises 1–4.

-30 -23 14 -21 -32 11 -23 28 103 -19 -5 -46

Statistics Find the mean, median, standard deviation, and range. Are the results statistics or parameters?

Critical Thinking: What does the survey tell us? Surveys have become an integral part of our lives. Because it is so important that every citizen has the ability to interpret survey results, surveys are the focus of this project. The Pew Research Center recently conducted a survey of 1007 U.S. adults and found that 85% of those surveyed know what Twitter is. 

Analyzing the Data

Use the survey results to construct a 95% confidence interval estimate of the percentage of all adults who know what Twitter is.

Vitamin C and Aspirin A bottle contains a label stating that it contains Spring Valley pills with 500 mg of vitamin C, and another bottle contains a label stating that it contains Bayer pills with 325 mg of aspirin. When testing claims about the mean contents of the pills, which would have more serious implications: rejection of the Spring Valley vitamin C claim or rejection of the Bayer aspirin claim? Is it wise to use the same significance level for hypothesis tests about the mean amount of vitamin C and the mean amount of aspirin?

Estimates and Hypothesis Tests Data Set 3 “Body Temperatures” in Appendix B includes sample body temperatures. We could use methods of Chapter 7 for making an estimate, or we could use those values to test the common belief that the mean body temperature is 98.6°F. What is the difference between estimating and hypothesis testing?

A formal hypothesis test is to be conducted using the claim that the mean height of men is equal to 174.1 cm.

a. What is the null hypothesis, and how is it denoted? 

b. What is the alternative hypothesis, and how is it denoted?

c. What are the possible conclusions that can be made about the null hypothesis?

d. Is it possible to conclude that “there is sufficient evidence to support the claim that the mean height of men is equal to 174.1 cm”?

The Ericsson method is one of several methods claimed to increase the likelihood of a baby girl. In a clinical trial, results could be analysed with a formal hypothesis test with the alternative hypothesis of p>0.5, which corresponds to the claim that the method increases the likelihood of having a girl, so that the proportion of girls is greater than 0.5. If you have an interest in establishing the success of the method, which of the following P-values would you prefer: 0.999, 0.5, 0.95, 0.05, 0.01, and 0.001? Why?

Identifying \({H_0}\) and \({H_1}\) . In Exercises 5–8, do the following: 

a. Express the original claim in symbolic form. 

b. Identify the null and alternative hypotheses.

Online Data Claim: Most adults would erase all of their personal information online if they could. A GFI Software survey of 565 randomly selected adults showed that 59% of them would erase all of their personal information online if they could.

Identifying \({H_0}\) and \({H_1}\) . In Exercises 5–8, do the following: 

a. Express the original claim in symbolic form. 

b. Identify the null and alternative hypotheses.

Cell Phone Claim: Fewer than 95% of adults have a cell phone. In a Marist poll of 1128 adults, 87% said that they have a cell phone.

Identifying \({H_0}\) and \({H_1}\) . In Exercises 5–8, do the following: 

a. Express the original claim in symbolic form. 

b. Identify the null and alternative hypotheses.

Pulse Rates Claim: The mean pulse rate (in beats per minute, or bpm) of adult males is equal to 69 bpm. For the random sample of 153 adult males in Data Set 1 “Body Data” in Appendix B, the mean pulse rate is 69.6 bpm and the standard deviation is 11.3 bpm.

Identifying \({H_0}\) and \({H_1}\) . In Exercises 5–8, do the following: 

a. Express the original claim in symbolic form. 

b. Identify the null and alternative hypotheses.

Pulse Rates Claim: The standard deviation of pulse rates of adult males is more than 11 bpm. For the random sample of 153 adult males in Data Set 1 “Body Data” in Appendix B, the pulse rates have a standard deviation of 11.3 bpm.

In Exercises 9–12, refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favours heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favours heads (because it is easy to get 11 heads in 20 flips by chance with a fair coin).

Exercise 5 “Online Data”

In Exercises 9–12, refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favours heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favours heads (because it is easy to get 11 heads in 20 flips by chance with a fair coin).

Exercise 6 “Cell Phone”

In Exercises 9–12, refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favours heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favours heads (because it is easy to get 11 heads in 20 flips by chance with a fair coin).

Exercise 7 “Pulse Rates”

In Exercises 9–12, refer to the exercise identified. Make subjective estimates to decide whether results are significantly low or significantly high, then state a conclusion about the original claim. For example, if the claim is that a coin favours heads and sample results consist of 11 heads in 20 flips, conclude that there is not sufficient evidence to support the claim that the coin favours heads (because it is easy to get 11 heads in 20 flips by chance with a fair coin).

Exercise 8 “Pulse Rates”

Test Statistics. In Exercises 13–16, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 on page 362 to select the correct expression for evaluating the test statistic.) 

Exercise 5 “Online Data”

Test Statistics. In Exercises 13–16, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 on page 362 to select the correct expression for evaluating the test statistic.) 

Exercise 6 “Cell Phone”

Test Statistics. In Exercises 13–16, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 on page 362 to select the correct expression for evaluating the test statistic.) 

Exercise 7 “Pulse Rates”

Test Statistics. In Exercises 13–16, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 on page 362 to select the correct expression for evaluating the test statistic.) 

16. Exercise 8 “Pulse Rates”

P-Values. In Exercises 17–20, do the following: 

a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed. 

b. Find the P-value. (See Figure 8-3 on page 364.) 

c. Using a significance level of \(\alpha \) = 0.05, should we reject \({H_0}\)or should we fail to reject \({H_0}\)?

The test statistic of z = 1.00 is obtained when testing the claim that \(p > 0.3\)

In Exercises 1–3, refer to the accompanying screen display that results from the Verizon airport data speeds (Mbps) from Data Set 32 “Airport Data Speeds” in Appendix B. The confidence level of 95% was used

Airport Data Speeds Refer to the accompanying screen display.

a.Express the confidence interval in the format that uses the “less than” symbol. Given that the original listed data use one decimal place, round the confidence interval limits accordingly.

b.Identify the best point estimate of \(\mu \)and the margin of error.

c.In constructing the confidence interval estimate of \(\mu \), why is it not necessary to confirm thatthe sample data appear to be from a population with a normal distribution?

In Exercises 1–3, refer to the accompanying screen display that results from the Verizon airport data speeds (Mbps) from Data Set 32 “Airport Data Speeds” in Appendix B. The confidence level of 95% was used. 

Degrees of Freedom

a. What is the number of degrees of freedom that should be used for finding the critical value \({t_{\frac{\alpha }{2}}}\)?

b. Find the critical value \({t_{\frac{\alpha }{2}}}\) corresponding to a 95% confidence level.

c. Give a brief general description of the number of degrees of freedom.

In Exercises 1–3, refer to the accompanying screen display that results from the Verizon airport data speeds (Mbps) from Data Set 32 “Airport Data Speeds” in Appendix B. The confidence level of 95% was used.

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.

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 \({t_{\frac{\alpha }{2}}}\) , (b) find the critical value \({z_{\frac{\alpha }{2}}}\), or (c) state that neither the normal distribution nor the t distribution applies.

Miami Heat Salaries Confidence level is 95%, \(\sigma \) is not known, and the normal quantile plot of the 17 salaries (thousands of dollars) of Miami Heat basketball players is as shown.

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   \({t_{{\alpha  \mathord{\left/

 {\vphantom {\alpha  2}} \right.

 \kern-\nulldelimiterspace} 2}}}\) ,(b) find the critical value \({z_{{\alpha  \mathord{\left/

 {\vphantom {\alpha  2}} \right.

 \kern-\nulldelimiterspace} 2}}}\) ,or (c) state that neither the normal distribution nor the  t distribution applies.

Denver Bronco Salaries confidence level is 90%,\(\sigma \) is not known, and the histogram of  61 player salaries (thousands of dollars) is as shown.

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   \({t_{{\alpha  \mathord{\left/

 {\vphantom {\alpha  2}} \right.

 \kern-\nulldelimiterspace} 2}}}\) ,(b) find the critical value \({z_{{\alpha  \mathord{\left/

 {\vphantom {\alpha  2}} \right.

 \kern-\nulldelimiterspace} 2}}}\) ,or (c) state that neither the normal distribution nor the  t distribution applies.

 Denver Bronco Salaries confidence level is 99%,\(\sigma  = 3342\)thousand dollars, and the histogram of 61 player salaries (thousands of dollars) is shown in Exercise 6.

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   \({t_{{\alpha  \mathord{\left/

 {\vphantom {\alpha  2}} \right.

 \kern-\nulldelimiterspace} 2}}}\) ,(b) find the critical value \({z_{{\alpha  \mathord{\left/

 {\vphantom {\alpha  2}} \right.

 \kern-\nulldelimiterspace} 2}}}\) ,or (c) state that neither the normal distribution nor the  t distribution applies.

Birth Weights Here are summary statistics for randomly selected weights of newborn girls: \({\bf{n}}\,{\bf{ = }}\,{\bf{205,\bar x}}\,{\bf{ = 30}}{\bf{.4}}\,{\bf{hg,s}}\,{\bf{ = 7}}{\bf{.1}}\,{\bf{hg}}\) (based on Data Set 4 “Births” in Appendix B). The confidence level is 95%.

Confidence Intervals. In Exercises 9–24, construct the confidence interval estimate of the mean.

Birth Weights of Girls Use these summary statistics given in Exercise 8:

\({\bf{n}}\,{\bf{ = }}\,{\bf{205,\bar x}}\,{\bf{ = 30}}{\bf{.4}}\,{\bf{hg,s}}\,{\bf{ = 7}}{\bf{.1}}\,{\bf{hg}}\). Use a 95% confidence level. Are the results very different from those found in Example 2 with only 15 sample values?

Birth Weights of Boys Use these summary statistics for birth weights of 195 boys: \({\bf{\bar x}}\,{\bf{ = 32}}{\bf{.7}}\,{\bf{hg}}\),\({\bf{s}}\,{\bf{ = 6}}{\bf{.6}}\,{\bf{hg}}\). (based on Data Set 4 “Births” in Appendix B). Use a 95%  confidencelevel. Are the results very different from those found in Exercise 9? Does it appear that boys and girls have very different birth weights?

Mean Body Temperature Data Set 3 “Body Temperatures” in Appendix B includes a sample of 106 body temperatures having a mean of 98.20°F and a standard deviation of 0.62°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°F is the mean body temperature?

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?

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?

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?

Genes Samples of DNA are collected, and the four DNA bases of A, G, C, and T are codedas 1, 2, 3, and 4, respectively. The results are listed below. Construct a 95% confidence intervalestimate of the mean. What is the practical use of the confidence interval?

                                                         2 2 1 4 3 3 3 3 4 1

Confidence Intervals. In Exercises 9–24, construct the confidence interval estimate of the mean. 

Arsenic in Rice Listed below are amounts of arsenic (μg, or micrograms, per serving) in samples of brown rice from California (based on data from the Food and Drug Administration). Use a 90% confidence level. The Food and Drug Administration also measured amounts of arsenic in samples of brown rice from Arkansas. Can the confidence interval be used to describe arsenic levels in Arkansas? 

5.4 5.6 8.4 7.3 4.5 7.5 1.5 5.5 9.1 8.7

Confidence Intervals. In Exercises 9–24, construct the confidence interval estimate of the mean.

In a study of speed dating conducted at Columbia University, male subjects were asked to rate the attractiveness of their female dates, and a sample of the results is listed below (1 = not attractive; 10 = extremely attractive). Use a 99% confidence level. What do the results tell us about the mean attractiveness ratings made by the population of all adult females?

7 8 2 10 6 5 7 8 8 9 5 9

Confidence Intervals. In Exercises 9–24, construct the confidence interval estimate of the mean.

In a study of speed dating conducted at Columbia University, female subjects were asked to rate the attractiveness of their male dates, and a sample of the results is listed below (1 = not attractive; 10 = extremely attractive). Use a 99% confidence level. Can the result be used to estimate the mean amount of attractiveness of the population of all adult males?

5 8 3 8 6 10 3 7 9 8 5 5 6 8 8 7 3 5 5 6 8 7 8 8 8 7

Confidence Intervals. In Exercises 9–24, construct the confidence interval estimate of the mean. In Exercises 2-19, Mercury in Sushi An FDA guideline is that themercury in fish should be below 1 part per million (ppm). Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different stores in New York City. The study was sponsored by the New YorkTimes, and the stores (in order) are D’Agostino, Eli’s Manhattan, Fairway, Food Emporium, Gourmet Garage, Grace’s Marketplace, and Whole Foods. Construct a 98% confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna sushi?

0.56 0.75 0.10 0.95 1.25 0.54 0.88

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 the data from the National Center for Education Statistics). Construct a 95% confidence interval estimate of the mean time for all college students to earn bachelor’s degrees. Does it appear that college students typically earn bachelor’s degrees in four years? Is there anything about the data that would suggest that the confidence interval might not be good result?

                   4   4    4   4   4    4   4.5   4.5    4.5  4.5  4.5   4.5  

                   6   6    8   9   9   13  13  15

Celebrity Net Worth Listed below are the amounts of net worth (in millions of dollars) of these ten wealthiest celebrities:  Tom Cruise, Will Smith, Robert De Niro, Drew Carey, George Clooney, John Travolta, Samuel L. Jackson, Larry King, Demi Moore, and Bruce Willis. Construct a 98% confidence interval. What does the result tell us about the population of all celebrities? Do the data appear to be from a normally distributed population as required?

              250  200   185  165  160   160  150  150   150  150

Caffeine in Soft Drinks Listed below are measured amounts of caffeine (mg per 12oz of drink) obtained in one can from each of 20 brands (7UP, A&W root Beer, Cherry Coke, …TaB). Use a confidence interval 99%. Does the confidence interval give us good information about the population of all cans of the same 20 brands that are consumed? Does the sample appear to be from a normally distributed population? If not, how are the results affected?

                   0  0   34  34  34   45  41  51   55  36  47   41  0  0   53  54  38   0  41  47

Confidence Intervals. In Exercises 9–24, construct the confidence interval estimate of the mean.

Student Evaluations Listed below are student evaluation ratings of courses, where a rating of 5 is for “excellent.” The ratings were obtained at the University of Texas at Austin. (See Data Set 17 “Course Evaluations” in Appendix B.) Use a 90% confidence level. What does the confidence interval tell us about the population of college students in Texas?

3.8 3.0 4.0 4.8 3.0 4.2 3.5 4.7 4.4 4.2 4.3 3.8 3.3 4.0 3.8

Confidence Intervals. In Exercises 9–24, construct the confidence interval estimate of the mean.

Flight Arrivals Listed below are arrival delays (minutes) of randomly selected American Airlines flights from New York (JFK) to Los Angeles (LAX). Negative numbers correspond to flights that arrived before the scheduled arrival time. Use a 95% confidence interval. How good is the on-time performance? 

                        -5 -32 -13 -9 -19 49 -30 -23 14 -21 -32 11

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

 Mean IQ of College Professors 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 college professors. We want to be 99% confident that our sample mean is within 4 IQ points of the true mean. The mean for this population is clearly greater than 100. The standard deviation for this population is less than 15 because it is a group with less variation than a group randomly selected from the general population; therefore, if we use \(\sigma  = 15\) we are being conservative by using a value that will make the sample size at least as large as necessary. Assume then that \(\sigma  = 15\)and determine the required sample size. Does the sample size appear to be practical?

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

Mean IQ of Attorneys See the preceding exercise, in which we can assume that \(\sigma  = 15\) for the IQ scores. Attorneys are a group with IQ scores that vary less than the IQ scores of the general population. Find the sample size needed to estimate the mean IQ of attorneys, given that we want 98% confidence that the sample mean is within 3 IQ points of the population mean. Does the sample size appear to be practical?

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

Mean Grade-Point Average Assume that all grade-point averages are to be standardized on a scale between 0 and 4. How many grade-point averages must be obtained so that the sample mean is within 0.01 of the population mean? Assume that a 95% confidence level is desired. If we use the range rule of thumb, we can estimate \(\sigma \)  to be,

  \(\begin{array}{c}\sigma  = \frac{{range}}{4}\\ = \frac{{4 - 0}}{4}\\ = 1\end{array}\)

 Does the sample size seem practical?

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

Mean Weight of Male Statistics Students Data Set 1 “Body Data” in Appendix B includes weights of 153 randomly selected adult males, and those weights have a standard deviation of 17.65 kg. Because it is reasonable to assume that weights of male statistics students have less variation than weights of the population of adult males, let \(\sigma  = 17.65\,kg\). How many male statistics students must be weighed in order to estimate the mean weight of all male statistics students? Assume that we want 90% confidence that the sample mean is within 1.5 kg of the population mean. Does it seem reasonable to assume that weights of male statistics students have less variation than weights of the population of adult males?

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

Mean Age of Female Statistics Students Data Set 1 “Body Data” in Appendix B  includes ages of 147 randomly selected adult females, and those ages have a standard deviation of 17.7 years. Assume that ages of female statistics students have less variation than ages of females in the general population, so let \(\sigma  = 17.7\)years for the sample size calculation. How many female statistics student ages must be obtained in order to estimate the mean age of all female statistics students? Assume that we want 95% confidence that the sample mean is within one-half year of the population mean. Does it seem reasonable to assume that ages of female statistics students have less variation than ages of females in the general population?

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

Mean Pulse Rate of Females Data Set 1 “Body Data” in Appendix B includes pulse rates of 147 randomly selected adult females, and those pulse rates vary from a low of 36 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult females. Assume that we want 99% confidence that the sample mean is within 2 bpm of the population mean.

a. Find the sample size using the range rule of thumb to estimate \(\sigma \).

b. Assume that \(\sigma  = 12.5\,\,bpm\) based on the value of \(s = 12.5\,\,bpm\)for the sample of 147 female pulse rates.

c. Compare the results from parts (a) and (b). Which result is likely to be better?