Trimmed Mean Another measure of central tendency is the

Problem 1E What does it mean if a statistic is resistant?

Problem 2E In the 2009 Current Population Survey conducted by the U.S. Census Bureau, two average household incomes were reported: $49,777 and $67,976. One of these averages is the mean and the other is the median. Which is the mean? Support your answer.

Problem 3E The U.S. Department of Housing and Urban Development (HUD) uses the median to report the average price of a home in the United States. Why do you think HUD uses the median?

Problem 4E A histogram of a set of data indicates that the distribution of the data is skewed right. Which measure of central tendency will likely be larger, the mean or the median? Why?

Problem 6E True or False : A data set will always have exactly one mode.

Problem 5E If a data set contains 10,000 values arranged in increasing order, where is the median located?

Problem 7E Find the population mean or sample mean as indicated. Sample: 20, 13, 4, 8, 10

Problem 9E Find the population mean or sample mean as indicated. Population: 3, 6, 10, 12, 14

Problem 8E Find the population mean or sample mean as indicated. Sample: 83, 65, 91, 87, 84

Problem 10E Find the population mean or sample mean as indicated. Population: 1, 19, 25, 15, 12, 16, 28, 13, 6

Problem 11E For Super Bowl XLIV, CBS television sold 55 ad slots for a total revenue of roughly $165.5 million. What was the mean price per ad slot?

Problem 12E The median for the given set of six ordered data values is 26.5. What is the missing value? 7 12 21_____41 50

Problem 13E Crash Test Results The Insurance Institute for Highway Safety crashed the 2010 Ford Fusion four times at 5 miles per hour. The costs of repair for each of the four crashes were $2529, $1889, $2610, $1073 Compute the mean, median, and mode cost of repair.

Problem 14E Cell Phone Use The following data represent the monthly cell phone bill for my wife’s phone for six randomly selected months. $35.34, $42.09, $39.43, $38.93, $43.39, $49.26 Compute the mean, median, and mode phone bill.

Problem 15E Concrete Mix A certain type of concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The strength of concrete is measured by pouring the mix into casting cylinders 6 inches in diameter and 12 inches tall. The concrete is allowed to set for 28 days. The concrete’s strength is then measured. The following data represent the strength of nine randomly selected casts (in psi). 3960, 4090, 3200, 3100, 2940, 3830, 4090, 4040, 3780 Compute the mean, median, and mode strength of the concrete (in psi).

Problem 16E Flight Time The following data represent the flight time (in minutes) of a random sample of seven flights from Las Vegas, Nevada, to Newark, New Jersey, on Continental Airlines. 282, 270, 260, 266, 257, 260, 267 Compute the mean, median, and mode flight time.

For each of the three histograms shown, determine whether the mean is greater than, less than, or approximately equal to the median. Justify your answer.

Problem 20E Reaction Time In an experiment conducted online at the University of Mississippi, study participants are asked to react to a stimulus. In one experiment, the participant must press a key upon seeing a blue screen. The time (in seconds) to press the key is measured. The same person is then asked to press a key upon seeing a red screen, again with the time to react measured. The table shows the results for six study participants. Participant Number Reaction Time to Blue Reaction Time to Red 1 0.582 0.408 2 0.481 0.407 3 0.841 0.542 4 0.267 0.402 5 0.685 0.456 6 0.450 0.533 (a) Determine the mean, median, and mode reaction time for both blue and red. ________________ (b) Does there appear to be a difference in the reaction time? What might account for any difference? How might this information be used? ________________ (c) Suppose the reaction time of 0.841 for blue was incorrectly recorded as 8.41. How does this affect the mean? the median? What property of the median does this illustrate?

Problem 19E pH in Water The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic; a pH greater than 7 is alkaline. The following data represent the pH in samples of bottled water and tap water. Tap 7.64 7.45 7.47 7.50 7.68 7.69 7.45 7.10 7.56 7.47 7.52 7.47 Bottled 5.15 5.09 5.26 5.20 5.02 5.23 5.28 5.26 5.13 5.26 5.21 5.24 (a) Determine the mean, median, and mode pH for each type of water. Comment on the differences between the two water types. ________________ (b) Suppose the pH of 7.10 in tap water was incorrectly recorded as 1.70. How does this affect the mean? the median? What property of the median does this illustrate?

Match the histograms shown to the summary statistics:

Problem 21E Pulse Rates The following data represent the pulse rates (beats per minute) of nine students enrolled in a section of Sullivan’s Introductory Statistics course. Treat the nine students as a population. Student Pulse Perpectual Bempah 76 Megan Brooks 60 Jeff Honeycutt 60 Clarice Jefferson 81 Crystal Kurtenbach 72 Janette Lantka 80 Kevin McCarthy 80 Tammy Ohm 68 Kathy Wojdyla 73 (a) Determine the population mean pulse. ________________ (b) Find three simple random samples of size 3 and determine the sample mean pulse of each sample. ________________ (c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Problem 23E Carbon Dioxide Emissions The given data represent the fossil-fuel carbon dioxide (CO2) emissions (in thousands of metric tons) of the top 10 emitters in 2007. Country Emissions Per Capita Emissions China 1,783,029 1.35 United States 1,591,756 5.20 India 439,695 0.39 Russia 419,241 2.95 Japan 342,117 2.71 Germany 214,872 2.61 Canada 151,988 4.61 United Kingdom 147,155 2.41 South Korea 137,257 2.82 Iran 135,257 1.88 (a) Determine the mean CO2 emissions of the top 10 countries. ________________ (b) Explain why the total emissions of a country is likely not the best gauge of CO2 emissions; instead, the emissions per capita (total emissions divided by population size) is the better gauge. ________________ (c) Determine the mean and median per capita CO2 emissions of the top 10 countries. Which measure would an environmentalist likely use to support the position that per capita CO2emissions are too high? Why?

Problem 24E Tour de Lance Lance Armstrong won the Tour de France seven consecutive years (1999–2005). The following table gives the winning times, distances, speeds, and margin of victory. Year Winning Time (h) Distance (km) Winning Speed (km/h) Winning Margin (min) 1999 91.538 3687 40.28 7.617 2000 92.552 3662 39.56 6.033 2001 86.291 3453 40.02 6.733 2002 82.087 3278 39.93 7.283 2003 83.687 3427 40.94 1.017 2004 83.601 3391 40.56 6.317 2005 86.251 3593 41.65 4.667 (a) Determine the mean and median of his winning times for the seven races. ________________ (b) Determine the mean and median of the distances for the seven races. ________________ (c) Determine the mean and median of his winning margin. ________________ (d) Determine the mean winning speed by finding the mean of the data values in the table. Next, compute the mean winning speed by finding the total of the seven distances and dividing by the total of the seven winning times. Finally, compute the mean winning speed by dividing the mean distance by the mean winning time. Do the three values agree or are there differences?

Problem 27E M&Ms The following data represent the weights (in grams) of a simple random sample of 50 M&M plain candies. 0.87 0.88 0.82 0.90 0.90 0.84 0.84 0.91 0.94 0.86 0.86 0.86 0.88 0.87 0.89 0.91 0.86 0.87 0.93 0.88 0.83 0.95 0.87 0.93 0.91 0.85 0.91 0.91 0.86 0.89 0.87 0.84 0.88 0.88 0.89 0.79 0.82 0.83 0.90 0.88 0.84 0.93 0.81 0.90 0.88 0.92 0.85 0.84 0.84 0.86 Determine the shape of the distribution of weights of M&Ms by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the weight of a plain M&M?

Journal Costs A histogram of the annual subscription cost (in dollars) for 26 biology journals is shown. The mean subscription cost is $1846 and the median subscription cost is $1142. Identify the shape of the distribution. Which measure of central tendency better describes the “center” of the distribution?

Problem 22E Travel Time The following data represent the travel time (in minutes) to school for nine students enrolled in Sullivan’s College Algebra course. Treat the nine students as a population. Student Travel Time Amanda 39 Amber 21 Tim 9 Mike 32 Nicole 30 Scot 45 Erica 11 Tiffany 12 Glenn 39 (a) Determine the population mean for travel time. ________________ (b) Find three simple random samples of size 4 and determine the sample mean for travel time of each sample. ________________ (c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Problem 28E Old Faithful We have all heard of the Old Faithful geyser in Yellowstone National Park. However, there is another, less famous, Old Faithful geyser in Calistoga, California. The following data represent the length of eruption (in seconds) for a random sample of eruptions of the California Old Faithful. 108 108 99 105 103 103 94 102 99 106 90 104 110 110 103 109 109 111 101 101 110 102 105 110 106 104 104 100 103 102 120 90 113 116 95 105 103 101 100 101 107 110 92 108 Determine the shape of the distribution of length of eruption by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the length of eruption?

Connection Time A histogram of the connection time, in seconds, to an Internet service provider for 30 randomly selected connections is shown. The mean connection time is 39.007 seconds and the median connection time is 39.065 seconds. Identify the shape of the distribution. Which measure of central tendency better describes the “center” of the distribution?

Problem 29E Hours Working A random sample of 25 college students was asked, “How many hours per week typically do you work outside the home?” Their responses were as follows: 0 0 15 20 30 40 30 20 35 35 28 15 20 25 25 30 5 0 30 24 28 30 35 15 15 Determine the shape of the distribution of hours worked by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes hours worked?

Problem 32E Hospital Admissions The following data represent the diagnosis of a random sample of 20 patients admitted to a hospital. Determine the mode diagnosis. Cancer Motor vehicle accident Congestive heart failure Gunshot wound Fall Gunshot wound Gunshot wound Motor vehicle accident Gunshot wound Assault Motor vehicle accident Gunshot wound Motor vehicle accident Motor vehicle accident Gunshot wound Motor vehicle accident Gunshot wound Motor vehicle accident Fall Gunshot wound

Problem 30E A Dealer’s Profit The following data represent the profit (in dollars) of a new-car dealer for a random sample of 40 sales. 781 1038 453 1446 3082 501 451 1826 1348 3001 1342 1889 580 0 2909 2883 480 1664 1064 2978 149 1291 507 261 540 543 87 798 673 2862 1692 1783 2186 398 526 730 2324 2823 1676 4148 Determine the shape of the distribution of new-car profit by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the profit?

Problem 31E Political Views A sample of 30 registered voters was surveyed in which the respondents were asked, “Do you consider your political views to be conservative, moderate, or liberal?” The results of the survey are shown in the table. Liberal Conservative Moderate Moderate Liberal Moderate Liberal Moderate Conservative Moderate Conservative Moderate Moderate Moderate Liberal Liberal Moderate Liberal Conservative Moderate Moderate Liberal Conservative Liberal Liberal Conservative Liberal Conservative Moderate Conservative (a) Determine the mode political view. ________________ (b) Do you think it would be a good idea to rotate the choices conservative, moderate, or liberal in the question? Why?

Problem 33E Resistance and Sample Size Each of the following three data sets represents the IQ scores of a random sample of adults. IQ scores are known to have a mean and median of 100. For each data set, determine the mean and median. For each data set recalculate the mean and median, assuming that the individual whose IQ is 106 is accidentally recorded as 160. For each sample size, state what happens to the mean and the median. Comment on the role the number of observations plays in resistance. SAMPLE OF SIZE 5 106 92 98 103 100 SAMPLE OF SIZE 12 106 92 98 103 100 102 98 124 83 70 108 121 SAMPLE OF SIZE 30 106 92 98 103 100 102 98 124 83 70 108 121 102 87 121 107 97 114 140 93 130 72 81 90 103 97 89 98 88 103

Problem 34E Mr. Zuro finds the mean height of all 14 students in his statistics class to be 68.0 inches. Just as Mr. Zuro finishes explaining how to get the mean, Danielle walks in late. Danielle is 65 inches tall. What is the mean height of the 15 students in the class?

Problem 35E Missing Exam Grade A professor has recorded exam grades for 20 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 19 readable scores is 84, what is the value of the unreadable score?

Survival Rates Unfortunately, a friend of yours has been diagnosed with cancer. You obtain a histogram of the survival time (in months) of patients diagnosed with this form of cancer as shown in the figure. The median survival time for individuals with this form of cancer is 11 months, while the mean survival time is 69 months. What words of encouragement should you share with your friend from a statistical point of view?

Problem 37E Sullivan Survey Choose any quantitative variable from the Sullivan Survey. Draw a histogram of the data. Determine the mean and the median. Based on the mean, median, and histogram, describe the shape of the distribution. Which measure of central tendency better describes the “center” of the distribution?

Problem 41E Trimmed Mean Another measure of central tendency is the trimmed mean. It is computed by determining the mean of a data set after deleting the smallest and largest observed values. Compute the trimmed mean for the data in.Problem Is the trimmed mean resistant? Explain. Problem M&Ms The following data represent the weights (in grams) of a simple random sample of 50 M&M plain candies. 0.87 0.88 0.82 0.90 0.90 0.84 0.84 0.91 0.94 0.86 0.86 0.86 0.88 0.87 0.89 0.91 0.86 0.87 0.93 0.88 0.83 0.95 0.87 0.93 0.91 0.85 0.91 0.91 0.86 0.89 0.87 0.84 0.88 0.88 0.89 0.79 0.82 0.83 0.90 0.88 0.84 0.93 0.81 0.90 0.88 0.92 0.85 0.84 0.84 0.86 Determine the shape of the distribution of weights of M&Ms by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the weight of a plain M&M?

Problem 38E Sullivan Survey Choose any quantitative variable from the Sullivan Survey. Now choose a qualitative variable, such as gender or political philosophy. Determine the mean and the median by the qualitative variable chosen. For example, if you chose gender as the qualitative variable, determine the mean and median by gender. Does there appear to be any difference in the measure of central tendency for each level of the qualitative variable?

Problem 39E Linear Transformations Benjamin owns a small Internet business. Besides himself, he employs nine other people. The salaries earned by the employees are given next in thousands of dollars (Benjamin’s salary is the largest, of course): 30, 30, 45, 50, 50, 50, 55, 55, 60, 75 (a) Determine the mean, median, and mode for salary. ________________ (b) Business has been good! As a result, Benjamin has a total of $25,000 in bonus pay to distribute to his employees. One option for distributing bonuses is to give each employee (including himself) $2500. Add the bonuses under this plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals? ________________ (c) As a second option, Benjamin can give each employee a bonus of 5% of his or her original salary. Add the bonuses under this second plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals? ________________ (d) As a third option, Benjamin decides not to give his employees a bonus at all. Instead, he keeps the $25,000 for himself. Use this plan to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?

Midrange The midrange is also a measure of central tendency. It is computed by adding the smallest and largest observed values of a data set and dividing the result by 2; that is, \(\text { Midrange }=\frac{\text { largest data value }+\text { smallest data value }}{2}\) Compute the midrange for the data in Problem 27. Is the midrange resistant? Explain. Equation Transcription: Text Transcription: Midrange =\frac{largest data value }+ { smallest data value }}{2}

Problem 40E Linear Transformations Use the five test scores of 65, 70, 71, 75, and 95 to answer the following questions: (a) Find the sample mean. ________________ (b) Find the median. ________________ (c) Which measure of central tendency best describes the typical test score? ________________ (d) Suppose the professor decides to curve the exam by adding 4 points to each test score. Compute the sample mean based on the adjusted scores. ________________ (e) Compare the unadjusted test score mean with the curved test score mean. What effect did adding 4 to each score have on the mean?

Problem 43E Putting It Together: Shape, Mean and MedianAs part of a semester project in a statistics course, Carlos surveyed a sample of 50 high school students and asked, “How many days in the past week have you consumed an alcoholic beverage?” The results of the survey are shown next. 0 0 1 4 1 1 1 5 1 3 0 1 0 1 0 4 0 1 0 1 0 0 0 0 2 0 0 0 0 0 1 0 2 0 0 0 1 2 1 1 2 0 1 0 1 3 1 1 0 3 (a) Is this data discrete or continuous? ________________ (b) Draw a histogram of the data and describe its shape. ________________ (c) Based on the shape of the histogram, do you expect the mean to be more than, equal to, or less than the median? ________________ (d) Determine the mean and the median. What does this tell you? ________________ (e) Determine the mode. ________________ (f ) Do you believe that Carlos’s survey suffers from sampling bias? Why?

Problem 44E FICO Scores The Fair Isaacs Corporation has devised a model that is used to determine creditworthiness of individuals, called a FICO score. FICO scores range in value from 300 to 850, with a higher score indicating a more credit-worthy individual. The distribution of FICO scores is skewed left with a median score of 723. (a) Do you think the mean FICO score is greater than, less than, or equal to 723? Justify your response. ________________ (b) What proportion of individuals have a FICO score above 723?

Problem 45E Why is the median resistant, but the mean is not?

Problem 46E A researcher with the Department of Energy wants to determine the mean natural gas bill of households throughout the United States. He knows the mean natural gas bill of households for each state, so he adds together these 50 values and divides by 50 to arrive at his estimate. Is this a valid approach? Why or why not?

Problem 48E You are negotiating a contract for the Players Association of the NBA. Which measure of central tendency will you use to support your claim that the average player’s salary needs to be increased? Why? As the chief negotiator for the owners, which measure would you use to refute the claim made by the Players Association?

Problem 49E In January 2011, the mean amount of money lost per visitor to a local riverboat casino was $135. Do you think the median was more than, less than, or equal to this amount? Why?

Problem 50E For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning. (a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2011 ________________ (b) Most popular major for students enrolled in a statistics course ________________ (c) Average test score when the scores are distributed symmetrically ________________ (d) Average test score when the scores are skewed right ________________ (e) Average income of a player in the National Football League ________________ (f ) Most requested song at a radio station

Problem 47E Net Worth According to the Statistical Abstract of the United States, the mean net worth of all households in the United States in 2007 was $556,300, while the median net worth was $120,300. (a) Which measure do you believe better describes the typical U.S. household’s net worth? Support your opinion. ________________ (b) What shape would you expect the distribution of net worth to have? Why? ________________ (c) What do you think causes the disparity in the two measures of central tendency?

What does it mean if a statistic is resistant?

In the 2009 Current Population Survey conducted by the U.S. Census Bureau, two average household incomes were reported: $49,777 and $67,976. One of these averages is the mean and the other is the median. Which is the mean? Support your answer

The U.S. Department of Housing and Urban Development (HUD) uses the median to report the average price of a home in the United States. Why do you think HUD uses the median?

A histogram of a set of data indicates that the distribution of the data is skewed right. Which measure of central tendency will likely be larger, the mean or the median? Why?

If a data set contains 10,000 values arranged in increasing order, where is the median located?

True or False : A data set will always have exactly one mode.

In Problems 710, find the population mean or sample mean as indicated Sample: 20, 13, 4, 8, 10

In Problems 710, find the population mean or sample mean as indicated Sample: 83, 65, 91, 87, 84

In Problems 710, find the population mean or sample mean as indicated Population: 3, 6, 10, 12, 14

In Problems 710, find the population mean or sample mean as indicated Population: 1, 19, 25, 15, 12, 16, 28, 13, 6

For Super Bowl XLIV, CBS television sold 55 ad slots for a total revenue of roughly $165.5 million. What was the mean price per ad slot?

The median for the given set of six ordered data values is 26.5. What is the missing value? 7 12 21 41 50

Crash Test Results The Insurance Institute for Highway Safety crashed the 2010 Ford Fusion four times at 5 miles per hour. The costs of repair for each of the four crashes were $2529, $1889, $2610, $1073 Compute the mean, median, and mode cost of repair.

Cell Phone Use The following data represent the monthly cell phone bill for my wifes phone for six randomly selected months. $35.34, $42.09, $39.43, $38.93, $43.39, $49.26 Compute the mean, median, and mode phone bill.

Concrete Mix A certain type of concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The strength of concrete is measured by pouring the mix into casting cylinders 6 inches in diameter and 12 inches tall. The concrete is allowed to set for 28 days. The concretes strength is then measured. The following data represent the strength of nine randomly selected casts (in psi). 3960, 4090, 3200, 3100, 2940, 3830, 4090, 4040, 3780 Compute the mean, median, and mode strength of the concrete (in psi).

Flight Time The following data represent the flight time (in minutes) of a random sample of seven flights from Las Vegas, Nevada, to Newark, New Jersey, on Continental Airlines. 282, 270, 260, 266, 257, 260, 267 Compute the mean, median, and mode flight time

For each of the three histograms shown, determine whether the mean is greater than, less than, or approximately equal to the median. Justify your answer. 0 9 15 21 27 33 3 6 12 18 24 30 (a) Frequency 20 15 10 5 0 0 4 8 12 16 20 (b) Frequency 9 8 7 6 5 4 3 2 1 0 0 5 15 25 10 20 30

Match the histograms shown to the summary statistics: Mean Median I 42 42 II 31 36 III 31 26 IV 31 32 5 15 25 35 45 55 (a) Frequency 15 10 5 0 10 10 30 50 70 90 (b) Frequency 9 8 7 6 5 4 3 2 1 0 7.5 7.5 22.5 37.5 52.5 67.5 (c) Frequency 10 5 0 14 26 38 50 62 7

pH in Water The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic; a pH greater than 7 is alkaline. The following data represent the pH in samples of bottled water and tap water. Tap 7.64 7.45 7.47 7.50 7.68 7.69 7.45 7.10 7.56 7.47 7.52 7.47 Bottled 5.15 5.09 5.26 5.20 5.02 5.23 5.28 5.26 5.13 5.26 5.21 5.24 Source: Emily McCarney, student at Joliet Junior College (a) Determine the mean, median, and mode pH for each type of water. Comment on the differences between the two water types. (b) Suppose the pH of 7.10 in tap water was incorrectly recorded as 1.70. How does this affect the mean? the median? What property of the median does this illustrate?

Reaction Time In an experiment conducted online at the University of Mississippi, study participants are asked to react to a stimulus. In one experiment, the participant must press a key upon seeing a blue screen. The time (in seconds) to press the key is measured. The same person is then asked to press a key upon seeing a red screen, again with the time to react measured. The table shows the results for six study participants. Participant Number Reaction Time to Blue Reaction Time to Red 1 0.582 0.408 2 0.481 0.407 3 0.841 0.542 4 0.267 0.402 5 0.685 0.456 6 0.450 0.533 Source: PsychExperiments at the University of Mississippi (www.olemiss.edu/psychexps) (a) Determine the mean, median, and mode reaction time for both blue and red. (b) Does there appear to be a difference in the reaction time? What might account for any difference? How might this information be used? (c) Suppose the reaction time of 0.841 for blue was incorrectly recorded as 8.41. How does this affect the mean? the median? What property of the median does this illustrate?

Pulse Rates The following data represent the pulse rates (beats per minute) of nine students enrolled in a section of Sullivans Introductory Statistics course. Treat the nine students as a population.(a) Determine the population mean pulse. (b) Find three simple random samples of size 3 and determine the sample mean pulse of each sample. (c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Travel Time The following data represent the travel time (in minutes) to school for nine students enrolled in Sullivans College Algebra course. Treat the nine students as a population. Student Travel Time Student Travel Time Amanda 39 Scot 45 Amber 21 Erica 11 Tim 9 Tiffany 12 Mike 32 Glenn 39 Nicole 30 (a) Determine the population mean for travel time. (b) Find three simple random samples of size 4 and determine the sample mean for travel time of each sample. (c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Carbon Dioxide Emissions The given data represent the fossil-fuel carbon dioxide (CO2) emissions (in thousands of metric tons) of the top 10 emitters in 2007. Country Emissions Per Capita Emissions China 1,783,029 1.35 United States 1,591,756 5.20 India 439,695 0.39 Russia 419,241 2.95 Japan 342,117 2.71 Germany 214,872 2.61 Canada 151,988 4.61 United Kingdom 147,155 2.41 South Korea 137,257 2.82 Iran 135,257 1.88 Source: Carbon Dioxide Information Analysis Center (a) Determine the mean CO2 emissions of the top 10 countries. (b) Explain why the total emissions of a country is likely not the best gauge of CO2 emissions; instead, the emissions per capita (total emissions divided by population size) is the better gauge. (c) Determine the mean and median per capita CO2 emissions of the top 10 countries. Which measure would an environmentalist likely use to support the position that per capita CO2 emissions are too high? Why?

Tour de Lance Lance Armstrong won the Tour de France seven consecutive years (19992005). The following table gives the winning times, distances, speeds, and margin of victory. Year Winning Time (h) Distance (km) Winning Speed (km/h) Winning Margin (min) 1999 91.538 3687 40.28 7.617 2000 92.552 3662 39.56 6.033 2001 86.291 3453 40.02 6.733 2002 82.087 3278 39.93 7.283 2003 83.687 3427 40.94 1.017 2004 83.601 3391 40.56 6.317 2005 86.251 3593 41.65 4.667 Source: cyclingnews.com (a) Determine the mean and median of his winning times for the seven races. (b) Determine the mean and median of the distances for the seven races. (c) Determine the mean and median of his winning margin. (d) Determine the mean winning speed by finding the mean of the data values in the table. Next, compute the mean winning speed by finding the total of the seven distances and dividing by the total of the seven winning times. Finally, compute the mean winning speed by dividing the mean distance by the mean winning time. Do the three values agree or are there differences?

Connection Time A histogram of the connection time, in seconds, to an Internet service provider for 30 randomly selected connections is shown. The mean connection time is 39.007 seconds and the median connection time is 39.065 seconds. Identify the shape of the distribution. Which measure of central tendency better describes the center of the distribution? 0 1 2 3 4 5 6 7 Frequency 36 38 40 42 44 Time (seconds) Connection Time Source: Nicole Spreitzer, student at Joliet Junior College

. Journal Costs A histogram of the annual subscription cost (in dollars) for 26 biology journals is shown. The mean subscription cost is $1846 and the median subscription cost is $1142. Identify the shape of the distribution. Which measure of central tendency better describes the center of the distribution? 0 2 4 6 8 10 12 14 Frequency 0 2000 4000 6000 8000 Cost (dollars) Cost of Biology Journals Source: Carol Wesolowski, student at Joliet Junior College

M&Ms The following data represent the weights (in grams) of a simple random sample of 50 M&M plain candies. 0.87 0.88 0.82 0.90 0.90 0.84 0.84 0.91 0.94 0.86 0.86 0.86 0.88 0.87 0.89 0.91 0.86 0.87 0.93 0.88 0.83 0.95 0.87 0.93 0.91 0.85 0.91 0.91 0.86 0.89 0.87 0.84 0.88 0.88 0.89 0.79 0.82 0.83 0.90 0.88 0.84 0.93 0.81 0.90 0.88 0.92 0.85 0.84 0.84 0.86 Source: Michael Sullivan Determine the shape of the distribution of weights of M&Ms by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the weight of a plain M&M?

Old Faithful We have all heard of the Old Faithful geyser in Yellowstone National Park. However, there is another, less famous, Old Faithful geyser in Calistoga, California. The following data represent the length of eruption (in seconds) for a random sample of eruptions of the California Old Faithful. 108 108 99 105 103 103 94 102 99 106 90 104 110 110 103 109 109 111 101 101 110 102 105 110 106 104 104 100 103 102 120 90 113 116 95 105 103 101 100 101 107 110 92 108 Source: Ladonna Hansen, Park Curator Determine the shape of the distribution of length of eruption by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the length of eruption?

Hours Working A random sample of 25 college students was asked, How many hours per week typically do you work outside the home? Their responses were as follows: 0 0 15 20 30 40 30 20 35 35 28 15 20 25 25 30 5 0 30 24 28 30 35 15 15 Determine the shape of the distribution of hours worked by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes hours worked?

A Dealers Profit The following data represent the profit (in dollars) of a new-car dealer for a random sample of 40 sales. 781 1038 453 1446 3082 501 451 1826 1348 3001 1342 1889 580 0 2909 2883 480 1664 1064 2978 149 1291 507 261 540 543 87 798 673 2862 1692 1783 2186 398 526 730 2324 2823 1676 4148 Source: Ashley Hudson, student at Joliet Junior College Determine the shape of the distribution of new-car profit by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the profit?

Political Views A sample of 30 registered voters was surveyed in which the respondents were asked, Do you consider your political views to be conservative, moderate, or liberal? The results of the survey are shown in the table. Liberal Conservative Moderate Moderate Liberal Moderate Liberal Moderate Conservative Moderate Conservative Moderate Moderate Moderate Liberal Liberal Moderate Liberal Conservative Moderate Moderate Liberal Conservative Liberal Liberal Conservative Liberal Conservative Moderate Conservative Source: Based on data from the General Social Survey (a) Determine the mode political view. (b) Do you think it would be a good idea to rotate the choices conservative, moderate, or liberal in the question? Why?

Hospital Admissions The following data represent the diagnosis of a random sample of 20 patients admitted to a hospital. Determine the mode diagnosis. Cancer Motor vehicle accident Congestive heart failure Gunshot wound Fall Gunshot wound Gunshot wound Motor vehicle accident Gunshot wound Assault Motor vehicle accident Gunshot wound Motor vehicle accident Motor vehicle accident Gunshot wound Motor vehicle accident Gunshot wound Motor vehicle accident Fall Gunshot wound Source: Tamela Ohm, student at Joliet Junior College

Resistance and Sample Size Each of the following three data sets represents the IQ scores of a random sample of adults. IQ scores are known to have a mean and median of 100. For each data set, determine the mean and median. For each data set recalculate the mean and median, assuming that the individual whose IQ is 106 is accidentally recorded as 160. For each sample size, state what happens to the mean and the median. Comment on the role the number of observations plays in resistance. SAMPLE OF SIZE 5 106 92 98 103 100 SAMPLE OF SIZE 12 106 92 98 103 100 102 98 124 83 70 108 121 SAMPLE OF SIZE 30 106 92 98 103 100 102 98 124 83 70 108 121 102 87 121 107 97 114 140 93 130 72 81 90 103 97 89 98 88 103

Mr. Zuro finds the mean height of all 14 students in his statistics class to be 68.0 inches. Just as Mr. Zuro finishes explaining how to get the mean, Danielle walks in late. Danielle is 65 inches tall. What is the mean height of the 15 students in the class?

Missing Exam Grade A professor has recorded exam grades for 20 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 19 readable scores is 84, what is the value of the unreadable score?

Survival Rates Unfortunately, a friend of yours has been diagnosed with cancer. You obtain a histogram of the survival time (in months) of patients diagnosed with this form of cancer as shown in the figure. The median survival time for individuals with this form of cancer is 11 months, while the mean survival time is 69 months. What words of encouragement should you share with your friend from a statistical point of view? 0 10 20 30 40 50 Percent 0 40 80 120 160 Survival Time (months)

Sullivan Survey Choose any quantitative variable from the Sullivan Survey. Draw a histogram of the data. Determine the mean and the median. Based on the mean, median, and histogram, describe the shape of the distribution. Which measure of central tendency better describes the center of the distribution?

Sullivan Survey Choose any quantitative variable from the Sullivan Survey. Now choose a qualitative variable, such as gender or political philosophy. Determine the mean and the median by the qualitative variable chosen. For example, if you chose gender as the qualitative variable, determine the mean and median by gender. Does there appear to be any difference in the measure of central tendency for each level of the qualitative variable?

Linear Transformations Benjamin owns a small Internet business. Besides himself, he employs nine other people. The salaries earned by the employees are given next in thousands of dollars (Benjamins salary is the largest, of course): 30, 30, 45, 50, 50, 50, 55, 55, 60, 75 (a) Determine the mean, median, and mode for salary. (b) Business has been good! As a result, Benjamin has a total of $25,000 in bonus pay to distribute to his employees. One option for distributing bonuses is to give each employee (including himself) $2500. Add the bonuses under this plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals? (c) As a second option, Benjamin can give each employee a bonus of 5% of his or her original salary. Add the bonuses under this second plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals? (d) As a third option, Benjamin decides not to give his employees a bonus at all. Instead, he keeps the $25,000 for himself. Use this plan to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?

Linear Transformations Use the five test scores of 65, 70, 71, 75, and 95 to answer the following questions: (a) Find the sample mean. (b) Find the median. (c) Which measure of central tendency best describes the typical test score?(d) Suppose the professor decides to curve the exam by adding 4 points to each test score. Compute the sample mean based on the adjusted scores. (e) Compare the unadjusted test score mean with the curved test score mean. What effect did adding 4 to each score have on the mean?

Trimmed Mean Another measure of central tendency is the trimmed mean. It is computed by determining the mean of a data set after deleting the smallest and largest observed values. Compute the trimmed mean for the data in Problem 27. Is the trimmed mean resistant? Explain.

Midrange The midrange is also a measure of central tendency. It is computed by adding the smallest and largest observed values of a data set and dividing the result by 2; that is, Midrange = largest data value + smallest data value 2 Compute the midrange for the data in Problem 27. Is the midrange resistant? Explain.

Putting It Together: Shape, Mean and Median As part of a semester project in a statistics course, Carlos surveyed a sample of 50 high school students and asked, How many days in the past week have you consumed an alcoholic beverage? The results of the survey are shown next. 0 0 1 4 1 1 1 5 13 0 1 0 1 0 4 0 1 01 0 0 0 0 2 0 0 0 00 1 0 2 0 0 0 1 2 11 2 0 1 0 1 3 1 1 03 (a) Is this data discrete or continuous? (b) Draw a histogram of the data and describe its shape. (c) Based on the shape of the histogram, do you expect the mean to be more than, equal to, or less than the median? (d) Determine the mean and the median. What does this tell you? (e) Determine the mode. (f ) Do you believe that Carloss survey suffers from sampling bias? Why?

FICO Scores The Fair Isaacs Corporation has devised a model that is used to determine creditworthiness of individuals,called a FICO score. FICO scores range in value from 300 to 850, with a higher score indicating a more credit-worthy individual. The distribution of FICO scores is skewed left with a median score of 723. (a) Do you think the mean FICO score is greater than, less than, or equal to 723? Justify your response. (b) What proportion of individuals have a FICO score above 723?

Why is the median resistant, but the mean is not?

A researcher with the Department of Energy wants to determine the mean natural gas bill of households throughout the United States. He knows the mean natural gas bill of households for each state, so he adds together these 50 values and divides by 50 to arrive at his estimate. Is this a valid approach? Why or why not?

Net Worth According to the Statistical Abstract of the United States, the mean net worth of all households in the United States in 2007 was $556,300, while the median net worth was $120,300. (a) Which measure do you believe better describes the typical U.S. households net worth? Support your opinion. (b) What shape would you expect the distribution of net worth to have? Why? (c) What do you think causes the disparity in the two measures of central tendency?

You are negotiating a contract for the Players Association of the NBA. Which measure of central tendency will you use to support your claim that the average players salary needs to be increased? Why? As the chief negotiator for the owners, which measure would you use to refute the claim made by the Players Association?

In January 2011, the mean amount of money lost per visitor to a local riverboat casino was $135. Do you think the median was more than, less than, or equal to this amount? Why?

For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning. (a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2011 (b) Most popular major for students enrolled in a statistics course (c) Average test score when the scores are distributed symmetrically (d) Average test score when the scores are skewed right (e) Average income of a player in the National Football League (f ) Most requested song at a radio station