. Pearsons Index of Skewness The English statistician Karl Pearson(18571936) introduced

Problem 46E Comparing Two Data Sets In Exercise, find the coefficient of variation for each of the two data sets. Then compare the results. SAT Scores Sample SAT scores for eight males and eight females are listed. Male SAT scores 1520 1750 2120 1380 1980 1650 1030 1710 Female SAT scores 1790 1510 1500 1950 2210 1870 1260 1590

Problem 1A Specify the lower limit to be 10 and the upper limit to be 20. Plot 10 points that have a mean of about 15 and a standard deviation of about 3. Write the estimates of the values of the points. Plot a point with a value of 15. What happens to the mean and standard deviation? Plot a point with a value of 20. What happens to the mean and standard deviation?

Employees Which state has the greatest number of manufacturing employees? Explain your reasoning.

Problem 1E Explain how to find the range of a data set. What is an advantage of using the range as a measure of variation? What is a disadvantage?

Problem 2A Specify the lower limit to be 30 and the upper limit to be 40. How can you plot eight points so that the points have the greatest possible standard deviation? Use the applet to plot the set of points and then use the formula for standard deviation to confirm the value given in the applet. How can you plot eight points so that the points have the least possible standard deviation? Explain

Mean Business Size Estimate the mean number of employees at a manufacturing business for each state. Use 1000 as the midpoint for \(" 500+"\) Equation Transcription: Text Transcription: " 500+"

Employees Which state has the greatest number of employees per manufacturing business? Explain your reasoning.

Problem 2E Explain how to find the deviation of an entry in a data set. What is the sum of all the deviations in any data set?

Problem 3E Why is the standard deviation used more frequently than the variance?

Standard Deviation Estimate the standard deviation for the number of employees at a manufacturing business for each state. Use 1000 as the midpoint for \(" 500+"\) Equation Transcription: Text Transcription: ''500+''

Standard Deviation Which state has the greatest standard deviation? Explain your reasoning.

Problem 4E Explain the relationship between variance and standard deviation. Can either of these measures be negative? Explain.

Distribution Describe the distribution of the number of employees at manufacturing businesses for each state.

Problem 6E Given a data set, how do you know whether to calculate ? or s?

Problem 5E Describe the difference between the calculation of population standard deviation and that of sample standard deviation.

Problem 7E Discuss the similarities and the differences between the Empirical Rule and Chebychev’s Theorem.

Problem 8E What must you know about a data set before you can use the Empirical Rule?

Graphical Reasoning In Exercises 9 and 10, find the range of the data set represented by the graph. Bride’s Age at First Marriage

Graphical Reasoning In Exercises 9 and 10, find the range of the data set represented by the graph.

Problem 12E In Exercise, compare your answer to part (a) with your answer to part (b). How do outliers affect the range of a data set? Exercise Archaeology The depths (in inches) at which 10 artifacts are found are listed. 20.7 24.8 30.5 26.2 36.0 34.3 30.3 29.5 27.0 38.5 (a) Find the range of the data set. ________________ (b) Change 38.5 to 60.5 and find the range of the new data set.

Problem 11E Archaeology The depths (in inches) at which 10 artifacts are found are listed. 20.7 24.8 30.5 26.2 36.0 34.3 30.3 29.5 27.0 38.5 (a) Find the range of the data set. ________________ (b) Change 38.5 to 60.5 and find the range of the new data set.

Problem 13E Finding Population Statistics In Exercise, find the range, mean, variance, and standard deviation of the population data set. Football Wins The numbers of regular season wins for each American Football Conference (AFC) team in 2012 (Source: National Football League) 13 10 12 11 7 8 6 6 10 7 12 4 6 5 2 2

Problem 15E Finding Sample Statistics In Exercise, find the range, mean, variance, and standard deviation of the sample data set. Ages of Shoppers The ages (in years) of a random sample of shoppers at a clothing outlet 16 18 19 17 14 15 17 17 17 16 19 22 24 14 16 14 17 16 14 18

Problem 14E Finding Population Statistics In Exercise, find the range, mean, variance, and standard deviation of the population data set. Weights of Presidents The weights (in pounds) of all U.S. presidents since 1952 (Source: The New York Times) 173 175 200 173 160 185 195 230 190 180

Problem 16E Find the range, mean, variance, and standard deviation of the sample data set. Pregnancy Durations. The durations (in days) of pregnancies for a random sample of mothers 277 291 295 280 268 278 291 277 282 279 296 285 269 293 267 281 286 269 264 299

Graphical Reasoning Both data sets shown in the stem-and-leaf plots have a mean of 165. One has a standard deviation of 16, and the other has a standard deviation of 24. By looking at the stem-and-leaf plots, which is which? Explain your reasoning. (a) (b)

Graphical Reasoning Both data sets shown in the histograms have a mean of 50. One has a standard deviation of 2.4, and the other has a standard deviation of 5. By looking at the histograms, which is which? Explain your reasoning. (a) (b)

Problem 19E Salary Offers You are applying for jobs at two companies. Company A offers starting salaries with ? = $31,000 and ? = $1000. Company B offers starting salaries with ? = $31,000 and ? = $5000. From which company are you more likely to get an offer of $33,000 or more? Explain your reasoning.

Problem 20E Salary Offers You are applying for jobs at two companies. Company C offers starting salaries with ? = $39,000 and ? = $4000. Company D offers starting salaries with ? = $39,000 and ? = $1500. From which company are you more likely to get an offer of $42,000 or more? Explain your reasoning.

Graphical Reasoning In Exercises 21–24, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ? (i) (ii) (iii) Equation Transcription: Text Transcription: Key: 1|5=15 Key: 1|5=15 Key: 1|5=15

Graphical Reasoning In Exercises 21–24, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ? (i) (ii) (iii)

Graphical Reasoning In Exercises 21–24, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ? (i) (ii) (iii)

Graphical Reasoning In Exercises 21–24, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ? (i) (ii) (iii)

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics. \(N=8 \) \(\mu=6\) \(\sigma=3\) Equation Transcription: Text Transcription: N=8 \mu=6 \sigma=3

Problem 25E Constructing Data Sets In Exercise, construct a data set that has the given statistics. N = 6 ? = 5 ? ? 2

Problem 27E Constructing Data Sets In Exercise, construct a data set that has the given statistics N = 8 ? = 6 ? ? 3

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics. \(n=6\) \(\bar{x}=7\) \(s \approx 2\) Equation Transcription: Text Transcription: n=6 \bar{x}=7 s \approx 2

Problem 29E Using the Empirical Rule In Exercise, use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 67 miles per hour, with a standard deviation of 4 miles per hour. Estimate the percent of vehicles whose speeds are between 63 miles per hour and 71 miles per hour. (Assume the data set has a bell-shaped distribution.)

Problem 30E Using the Empirical Rule In Exercise, use the Empirical Rule. The mean monthly utility bill for a sample of households in a city is $70, with a standard deviation of $8. Between what two values do about 95% of the data lie? (Assume the data set has a bell-shaped distribution.)

Problem 32E Using the Empirical Rule In Exercise, use the Empirical Rule. Use the sample statistics from Exercise 30 and assume the number of households in the sample is 40. (a) Estimate the number of households whose monthly utility bills are between $54 and $86. ________________ (b) In a sample of 20 additional households, about how many households would you expect to have monthly utility bills between $54 and $86? In Exercise, use the Empirical Rule. The mean monthly utility bill for a sample of households in a city is $70, with a standard deviation of $8. Between what two values do about 95% of the data lie? (Assume the data set has a bell-shaped distribution.)

Problem 33E Using the Empirical Rule In Exercise, use the Empirical Rule. The speeds for eight more vehicles are listed. Using the sample statistics from Exercise 29, determine which of the data entries are unusual. Are any of the data entries very unusual? Explain your reasoning. 70, 78, 62, 71, 65, 76, 82, 64 In Exercise, use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 67 miles per hour, with a standard deviation of 4 miles per hour. Estimate the percent of vehicles whose speeds are between 63 miles per hour and 71 miles per hour. (Assume the data set has a bell-shaped distribution.)

Problem 31E Using the Empirical Rule In Exercise, use the Empirical Rule. Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75. (a) Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour. ________________ (b) In a sample of 25 additional vehicles, about how many vehicles would you expect to have speeds between 63 miles per hour and 71 miles per hour? In Exercise, use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 67 miles per hour, with a standard deviation of 4 miles per hour. Estimate the percent of vehicles whose speeds are between 63 miles per hour and 71 miles per hour. (Assume the data set has a bell-shaped distribution.)

Problem 36E Chebychev’s Theorem Old Faithful is a famous geyser at Yellowstone National Park. From a sample with n = 32, the mean duration of Old Faithful’s eruptions is 3.32 minutes and the standard deviation is 1.09 minutes. Using Chebychev’s Theorem, determine at least how many of the eruptions lasted between 1.14 minutes and 5.5 minutes. (Source: Yellowstone National Park)

Problem 37E Chebychev’s Theorem The mean score on a European history exam is 88 points, with a standard deviation of 4 points. Apply Chebychev’s Theorem to the data using k = 2. Interpret the results.

Problem 35E Chebychev’s Theorem You are conducting a survey on the number of pets per household in your region. From a sample with n = 40, the mean number of pets per household is 2 pets and the standard deviation is 1 pet. Using Chebychev’s Theorem, determine at least how many of the households have 0 to 4 pets.

Problem 38E Chebychev’s Theorem The mean time in the finals for the women’s 800-meter freestyle at the 2012 Summer Olympics was 502.84 seconds, with a standard deviation of 4.68 seconds. Apply Chebychev’s Theorem to the data using k = 2. Interpret the results. (Adapted from International Olympic Committee)

Problem 34E Using the Empirical Rule In Exercise, use the Empirical Rule. The monthly utility bills for eight more households are listed. Using the sample statistics from Exercise 30, determine which of the data entries are unusual. Are any of the data entries very unusual? Explain your reasoning. $65, $52, $63, $83, $77, $98, $84, $70 In Exercise, use the Empirical Rule. The mean monthly utility bill for a sample of households in a city is $70, with a standard deviation of $8. Between what two values do about 95% of the data lie? (Assume the data set has a bell-shaped distribution.)

Calculating Using Grouped Data In Exercises 39–42, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. Cars per Household The results of a random sample of the number of cars per household in a region are shown in the histogram.

Calculating Using Grouped Data In Exercises 39–42, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. Amounts of Caffeine The amounts of caffeine in a sample of five-ounce servings of brewed coffee are shown in the histogram.

Calculating Using Grouped Data In Exercises 39–42, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. Weekly Study Hours The distribution of the numbers of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for \(\text { "30+ hours." }\) Equation Transcription: Text Transcription: ''30+ hours''

Problem 43E Comparing Two Data Sets In Exercise, find the coefficient of variation for each of the two data sets. Then compare the results. Annual Salaries Sample annual salaries (in thousands of dollars) for entry level accountants in Dallas and New York City are listed. Dallas 41.6 50.0 49.5 38.7 39.9 45.8 44.7 47.8 40.5 44.3 New York City 45.6 41.5 57.6 55.1 59.3 59.0 50.6 47.2 42.3 51.0

Calculating Using Grouped Data In Exercises 39–42, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. Household Income The distribution of the monthly household incomes of a random sample of households in a city is shown in the pie chart. Use \(\$ 10,999.50\) as the midpoint for “\(\$ 10,000\) or more.” Equation Transcription: Text Transcription: $10,999.50 $10,000

Problem 44E Comparing Two Data Sets In Exercise, find the coefficient of variation for each of the two data sets. Then compare the results. Annual Salaries Sample annual salaries (in thousands of dollars) for entry level electrical engineers in Boston and Chicago are listed. Boston 70.4 84.2 58.5 64.5 71.6 79.9 88.3 80.1 69.9 Chicago 69.4 71.5 65.4 59.9 70.9 68.5 62.9 70.1 60.9

Comparing Two Data Sets In Exercises 43– 48, find the coefficient of variation for each of the two data sets. Then compare the results. Ages and Heights The ages (in years) and heights (in inches) of all pitchers for the 2013 St. Louis Cardinals are listed. (Source: Major League Baseball)

. Explain how to find the range of a data set. What is an advantage of using the range as a measure of variation? What is a disadvantage?

Explain how to find the deviation of an entry in a data set. What is the sum of all the deviations in any data set?

Why is the standard deviation used more frequently than the variance?

Explain the relationship between variance and standard deviation. Can either of these measures be negative? Explain.

Describe the difference between the calculation of population standard deviation and that of sample standard deviation.

Given a data set, how do you know whether to calculate s or s?

Discuss the similarities and the differences between the Empirical Rule and Chebychevs Theorem.

What must you know about a data set before you can use the Empirical Rule?

Graphical Reasoning In Exercises 9 and 10, find the range of the data set represented by the graph Brides Age at First Marriage

Graphical Reasoning In Exercises 9 and 10, find the range of the data set represented by the graph

Archaeology The depths (in inches) at which 10 artifacts are found are listed. 20.7 24.8 30.5 26.2 36.0 34.3 30.3 29.5 27.0 38.5 (a) Find the range of the data set. (b) Change 38.5 to 60.5 and find the range of the new data set.

. In Exercise 11, compare your answer to part (a) with your answer to part (b). How do outliers affect the range of a data set?

Finding Population Statistics In Exercises 13 and 14, find the range, mean, variance, and standard deviation of the population data set.Football Wins The numbers of regular season wins for each American Football Conference (AFC) team in 2012 (Source: National Football League) 13 10 12 11 7 8 6 6 10 7 12 4 6 5 2 2

Weights of Presidents The weights (in pounds) of all U.S. presidents since 1952 (Source: The New York Times) 173 175 200 173 160 185 195 230 190 180

Finding Sample Statistics In Exercises 15 and 16, find the range, mean, variance, and standard deviation of the sample data set. Ages of Shoppers The ages (in years) of a random sample of shoppers at a clothing outlet 16 18 19 17 14 15 17 17 17 16 19 22 24 14 16 14 17 16 14 18

Finding Sample Statistics In Exercises 15 and 16, find the range, mean, variance, and standard deviation of the sample data setPregnancy Durations The durations (in days) of pregnancies for a random sample of mothers 277 291 295 280 268 278 291 277 282 279 296 285 269 293 267 281 286 269 264 299

Graphical Reasoning Both data sets shown in the stem-and-leaf plots have a mean of 165. One has a standard deviation of 16, and the other has a standard deviation of 24. By looking at the stem-and-leaf plots, which is which? Explain your reasoning. (a) 12 8 9 Key: 12 0 8 = 128 13 5 5 8 14 1 2 15 0 0 6 7 16 4 5 9 17 1 3 6 8 18 0 8 9 19 6 20 3 5 7 (b) 12 Key: 13 0 1 = 131 13 1 14 2 3 5 15 0 4 5 6 8 16 1 1 2 3 3 3 17 1 5 8 8 18 2 3 4 5 19 0 2 20

Graphical Reasoning Both data sets shown in the histograms have a mean of 50. One has a standard deviation of 2.4, and the other has a standard deviation of 5. By looking at the histograms, which is which? Explain your reasoning. (a) 20 15 10 5 42 45 48 51 54 57 60 Frequency Data entry (b) 20 15 10 5 42 45 48 51 54 57

Salary Offers You are applying for jobs at two companies. Company A offers starting salaries with m = $31,000 and s = $1000. Company B offers starting salaries with m = $31,000 and s = $5000. From which company are you more likely to get an offer of $33,000 or more? Explain your reasoning

Salary Offers You are applying for jobs at two companies. Company C offers starting salaries with m = $39,000 and s = $4000. Company D offers starting salaries with m = $39,000 and s = $1500. From which company are you more likely to get an offer of $42,000 or more? Explain your reasoning.

Graphical Reasoning In Exercises 2124, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ?

Graphical Reasoning In Exercises 2124, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ?

Graphical Reasoning In Exercises 2124, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ?

Graphical Reasoning In Exercises 2124, you are asked to compare three data sets. (a) Without calculating, determine which data set has the greatest sample standard deviation and which has the least sample standard deviation. Explain your reasoning. (b) How are the data sets the same? How do they differ?

Constructing Data Sets In Exercises 2528, construct a data set that has the given statistics.. N = 6 26. N = 8 m = 5 m = 6 s 2 s 3 27. n = 7 28. n = 6 x = 9 x = 7 s = 0 s 2

Constructing Data Sets In Exercises 2528, construct a data set that has the given statistics.. N = 6 26. N = 8 m = 5 m = 6 s 2 s 3 27. n = 7 28. n = 6 x = 9 x = 7 s = 0 s 2

Constructing Data Sets In Exercises 2528, construct a data set that has the given statistics.. N = 6 26. N = 8 m = 5 m = 6 s 2 s 3 27. n = 7 28. n = 6 x = 9 x = 7 s = 0 s 2

Constructing Data Sets In Exercises 2528, construct a data set that has the given statistics.. N = 6 26. N = 8 m = 5 m = 6 s 2 s 3 27. n = 7 28. n = 6 x = 9 x = 7 s = 0 s 2

Using the Empirical Rule In Exercises 2934, use the Empirical Rule.The mean speed of a sample of vehicles along a stretch of highway is 67 miles per hour, with a standard deviation of 4 miles per hour. Estimate the percent of vehicles whose speeds are between 63 miles per hour a

Using the Empirical Rule In Exercises 2934, use the Empirical Rule.The mean monthly utility bill for a sample of households in a city is $70, with a standard deviation of $8. Between what two values do about 95% of the data lie? (Assume the data set has a bell-shaped distribution.)

Using the Empirical Rule In Exercises 2934, use the Empirical Rule.Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75. (a) Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour. (b) In a sample of 25 additional vehicles, about how many vehicles would you expect to have speeds between 63 miles per hour and 71 miles per hour?

Using the Empirical Rule In Exercises 2934, use the Empirical Rule.Use the sample statistics from Exercise 30 and assume the number of households in the sample is 40. (a) Estimate the number of households whose monthly utility bills are between $54 and $86. (b) In a sample of 20 additional households, about how many households would you expect to have monthly utility bills between $54 and $86?

Using the Empirical Rule In Exercises 2934, use the Empirical Rule.The speeds for eight more vehicles are listed. Using the sample statistics from Exercise 29, determine which of the data entries are unusual. Are any of the data entries very unusual? Explain your reasoning. 70, 78, 62, 71, 65, 76, 82, 64

Using the Empirical Rule In Exercises 2934, use the Empirical Rule.The monthly utility bills for eight more households are listed. Using the sample statistics from Exercise 30, determine which of the data entries are unusual. Are any of the data entries very unusual? Explain your reasoning. $65, $52, $63, $83, $77, $98, $84, $70

Chebychevs Theorem You are conducting a survey on the number of pets per household in your region. From a sample with n = 40, the mean number of pets per household is 2 pets and the standard deviation is 1 pet. Using Chebychevs Theorem, determine at least how many of the households have 0 to 4 pets.

Chebychevs Theorem Old Faithful is a famous geyser at Yellowstone National Park. From a sample with n = 32, the mean duration of Old Faithfuls eruptions is 3.32 minutes and the standard deviation is 1.09 minutes. Using Chebychevs Theorem, determine at least how many of the eruptions lasted between 1.14 minutes and 5.5 minutes. (Source: Yellowstone National Park)

Chebychevs Theorem The mean score on a European history exam is 88 points, with a standard deviation of 4 points. Apply Chebychevs Theorem to the data using k = 2. Interpret the results.

Chebychevs Theorem The mean time in the finals for the womens 800-meter freestyle at the 2012 Summer Olympics was 502.84 seconds, with a standard deviation of 4.68 seconds. Apply Chebychevs Theorem to the data using k = 2. Interpret the results. (Adapted from International Olympic Committee

Calculating Using Grouped Data In Exercises 3942, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.. Cars per Household The results of a random sample of the number of cars per household in a region are shown in the histogram. Number of cars 5 10 15 20 25

Calculating Using Grouped Data In Exercises 3942, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.Amounts of Caffeine The amounts of caffeine in a sample of five-ounce servings of brewed coffee are shown in the histogram.

Calculating Using Grouped Data In Exercises 3942, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.1. Weekly Study Hours The distribution of the numbers of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for 30+ hours.

Calculating Using Grouped Data In Exercises 3942, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.Household Income The distribution of the monthly household incomes of a random sample of households in a city is shown in the pie chart. Use $10,999.50 as the midpoint for $10,000 or more.

Comparing Two Data Sets In Exercises 4348, find the coefficient of variation for each of the two data sets. Then compare the results.Annual Salaries Sample annual salaries (in thousands of dollars) for entry level accountants in Dallas and New York City are listed. Dallas 41.6 50.0 49.5 38.7 39.9 45.8 44.7 47.8 40.5 44.3 New York City 45.6 41.5 57.6 55.1 59.3 59.0 50.6 47.2 42.3 51.0

Comparing Two Data Sets In Exercises 4348, find the coefficient of variation for each of the two data sets. Then compare the results.Annual Salaries Sample annual salaries (in thousands of dollars) for entry level electrical engineers in Boston and Chicago are listed. Boston 70.4 84.2 58.5 64.5 71.6 79.9 88.3 80.1 69.9 Chicago 69.4 71.5 65.4 59.9 70.9 68.5 62.9 70.1 60.9

Comparing Two Data Sets In Exercises 4348, find the coefficient of variation for each of the two data sets. Then compare the results.Ages and Heights The ages (in years) and heights (in inches) of all pitchers for the 2013 St. Louis Cardinals are listed. (Source: Major League Baseball) Ages 24 29 37 24 26 25 24 32 22 29 23 31 Heights 72 76 73 73 77 76 72 74 75 75 74 79

Comparing Two Data Sets In Exercises 4348, find the coefficient of variation for each of the two data sets. Then compare the results.. SAT Scores Sample SAT scores for eight males and eight females are listed. Male SAT scores 1520 1750 2120 1380 1980 1650 1030 1710 Female SAT scores 1790 1510 1500 1950 2210 1870 1260 1590

Comparing Two Data Sets In Exercises 4348, find the coefficient of variation for each of the two data sets. Then compare the results.Batting Averages Sample batting averages for baseball players from two opposing teams are listed. Team A 0.295 0.310 0.325 0.272 0.256 0.297 0.320 0.384 0.235 0.297 Team B 0.223 0.312 0.256 0.300 0.238 0.299 0.204 0.226 0.292 0.260

Comparing Two Data Sets In Exercises 4348, find the coefficient of variation for each of the two data sets. Then compare the results.Ages and Weights The ages (in years) and weights (in pounds) of all wide receivers for the 2012 San Diego Chargers are listed. (Source: ESPN) Ages 25 24 24 31 25 28 26 30 22 Weights 215 217 190 225 192 215 185 210 220

Shortcut Formula You used SSx = (x - x)2 when calculating variance and standard deviation. An alternative formula that is sometimes more convenient for hand calculations is SSx = x2 - 1x22 n . You can find the sample variance by dividing the sum of squares by n - 1 and the sample standard deviation by finding the square root of the sample variance. (a) Use the shortcut formula to calculate the sample standard deviation for the data set in Exercise 15. (b) Compare your result with the sample standard deviation obtained in Exercise 15.

Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed. 42 36 48 51 39 39 42 36 48 33 39 42 45 (a) Find the sample mean and the sample standard deviation. (b) Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set. (c) To calculate the monthly salary, divide each original salary by 12. Find the sample mean and the sample standard deviation for the revised data set. (d) What can you conclude from the results of (a), (b), and (c)?

Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed. 40 35 49 53 38 39 40 37 49 34 38 43 47 (a) Find the sample mean and the sample standard deviation. (b) Each employee in the sample receives a $1000 raise. Find the sample mean and the sample standard deviation for the revised data set. (c) Each employee in the sample takes a pay cut of $2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set. (d) What can you conclude from the results of (a), (b), and (c)?

Mean Absolute Deviation Another useful measure of variation for a data set is the mean absolute deviation (MAD). It is calculated by the formula MAD = 0 x - x 0 n . (a) Find the mean absolute deviation of the data set in Exercise 15. Compare your result with the sample standard deviation obtained in Exercise 15. (b) Find the mean absolute deviation of the data set in Exercise 16. Compare your result with the sample standard deviation obtained in Exercise 16.

Chebychevs Theorem At least 99% of the data in any data set lie within how many standard deviations of the mean? Explain how you obtained your answer.

. Pearsons Index of Skewness The English statistician Karl Pearson (18571936) introduced a formula for the skewness of a distribution. P = 31x - median2 s Pearsons index of skewness Most distributions have an index of skewness between -3 and 3. When P 7 0, the data are skewed right. When P 6 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each. (a) x = 17, s = 2.3, median = 19 (b) x = 32, s = 5.1, median = 25 (c) x = 9.2, s = 1.8, median = 9.2 (d) x = 42, s = 6.0, median = 40