?Sullivan Survey Choose any quantitative variable from the SullivanStats Survey I at

The sum of the deviations about the mean always equals ________.

The standard deviation is used in conjunction with the ______ to numerically describe distributions that are bell shaped. The ______ measures the center of the distribution, while the standard deviation measures the ______ of the distribution.

True or False: When comparing two populations, the larger the standard deviation, the more dispersion the distribution has, provided that the variable of interest from the two populations has the same unit of measure.

True or False: Chebyshev’s Inequality applies to all distributions regardless of shape, but the Empirical Rule holds only for distributions that are bell shaped.

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Sample: 20, 13, 4, 8, 10

In Problems 5-10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Sample: 83, 65, 91, 87, 84

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Population: 3, 6, 10, 12, 14

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Population: 1, 19, 25, 15, 12, 16, 28, 13, 6

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Sample: 6, 52, 13, 49, 35, 25, 31, 29, 31, 29

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated. Population: 4, 10, 12, 12, 13, 21

Miles per Gallon The following data represent the miles per gallon for a 2013 Ford Fusion for six randomly selected vehicles. Compute the range, sample variance, and sample standard deviation miles per gallon. Source: www.fueleconomy.gov 34.0, 33.2, 37.0, 29.4, 23.6, 25.9

Exam Time The following data represent the amount of time (in minutes) a random sample of eight students took to complete the online portion of an exam in Sullivan’s Statistics course. Compute the range, sample variance, and sample standard deviation time. 60.5, 128.0, 84.6, 122.3, 78.9, 94.7, 85.9, 89.9

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 after it is allowed to set up for 28 days. The following data represent the strength of nine randomly selected casts. Compute the range and sample standard deviation for the strength of the concrete (in psi). 3960, 4090, 3200, 3100, 2940, 3830, 4090, 4040, 3780

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 United Airlines. Compute the range and sample standard deviation of flight time. 282, 270, 260, 266, 257, 260, 267

Which histogram depicts a higher standard deviation? Justify your answer.

Match the histograms in the next column to the summary statistics given.

Exam Scores The following data represent exam scores in a statistics class taught using traditional lecture and a class taught using a “flipped” classroom. The “flipped” classroom is one where the content is delivered via video and watched at home, while class time is used for homework and activities. (a) Which course has more dispersion in exam scores using the range as the measure of dispersion? (b) Which course has more dispersion in exam scores using the standard deviation as the measure of dispersion? (c) Suppose the score of 59.8 in the traditional course was incorrectly recorded as 598. How does this affect the range? the standard deviation? What property does this illustrate?

pH in Water The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic, while a pH greater than 7 is alkaline. The following data represent the pH in samples of bottled water and tap water. (a) Which type of water has more dispersion in pH using the range as the measure of dispersion? (b) Which type of water has more dispersion in pH using the standard deviation as the measure of dispersion?

Pulse Rates The data on the following page represent the pulse rates (beats per minute) of nine students enrolled in a section of Sullivan’s course in Introductory Statistics. Treat the nine students as a population. (a) Determine the population standard deviation. (b) Find three simple random samples of size 3, and determine the sample standard deviation of each sample. (c) Which samples underestimate the population standard deviation? Which overestimate the population standard deviation?

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. (a) Determine the population standard deviation. (b) Find three simple random samples of size 4, and determine the sample standard deviation of each sample. (c) Which samples underestimate the population standard deviation? Which overestimate the population standard deviation?

A Fish Story Ethan and Drew went on a 10-day fishing trip. The number of smallmouth bass caught and released by the two boys each day was as follows: (a) Find the population mean and the range for the number of smallmouth bass caught per day by each fisherman. Do these values indicate any differences between the two fishermen’s catches per day? Explain. (b) Draw a dot plot for Ethan. Draw a dot plot for Drew. Which fisherman seems more consistent? (c) Find the population standard deviation for the number of smallmouth bass caught per day by each fisherman. Do these values present a different story about the two fishermen’s catches per day? Which fisherman has the more consistent record? Explain. (d) Discuss limitations of the range as a measure of dispersion.

Soybean Yield The data in the next column represent the number of pods on a sample of soybean plants for two different plot types. Which plot type do you think is superior? Why?

The Empirical Rule The following data represent the weights (in grams) of a random sample of 50 M&M plain candies. a) Determine the sample standard deviation weight. Express your answer rounded to three decimal places. (b) On the basis of the histogram drawn in Section 3.1, Problem 25, comment on the appropriateness of using the Empirical Rule to make any general statements about the weights of M&Ms. (c) Use the Empirical Rule to determine the percentage of M&Ms with weights between 0.803 and 0.947 gram. Hint: = 0.875. (d) Determine the actual percentage of M&Ms that weigh between 0.803 and 0.947 gram, inclusive. (e) Use the Empirical Rule to determine the percentage of M&Ms with weights more than 0.911 gram. (f) Determine the actual percentage of M&Ms that weigh more than 0.911 gram.

The Empirical Rule The following data represent the length of eruption for a random sample of eruptions at the Old Faithful geyser in Calistoga, California. (a) Determine the sample standard deviation length of eruption. Express your answer rounded to the nearest whole number. (b) On the basis of the histogram drawn in Section 3.1, Problem 26, comment on the appropriateness of using the Empirical Rule to make any general statements about the length of eruptions. (c) Use the Empirical Rule to determine the percentage of eruptions that last between 92 and 116 seconds. Hint: = 104 (d) Determine the actual percentage of eruptions that last between 92 and 116 seconds, inclusive. (e) Use the Empirical Rule to determine the percentage of eruptions that last less than 98 seconds. (f) Determine the actual percentage of eruptions that last less than 98 seconds.

Which Car Would You Buy? Suppose that you are in the market to purchase a car. You have narrowed it down to two choices and will let gas mileage be the deciding factor. You decide to conduct a little experiment in which you put 10 gallons of gas in the car and drive it on a closed track until it runs out of gas. You conduct this experiment 15 times on each car and record the number of miles driven. Describe each data set. That is, determine the shape, center, and spread. Which car would you buy and why?

Which Investment Is Better? You have received a year-end bonus of $5000. You decide to invest the money in the stock market and have narrowed your investment options down to two mutual funds. The following data represent the historical quarterly rates of return of each mutual fund for the past 20 quarters (5 years). Describe each data set. That is, determine the shape, center, and spread. Which mutual fund would you invest in and why?

Rates of Returns of Stocks Stocks may be categorized by sectors. Go to www.pearsonhighered.com/sullivanstats and download 3_2_27 using the file format of your choice. The data represent the one-year rate of return (in percent) for a sample of consumer cyclical stocks and industrial stocks for the period December 2013 through November 2014. Note: Consumer cyclical stocks include names such as Starbucks and Home Depot. Industrial stocks include names such as 3M and FedEx. (a) Draw a relative frequency histogram for each sector using a lower class limit for the first class of -50 and a class width of 10. Which sector appears to have more dispersion? (b) Determine the mean and median rate of return for each sector. Which sector has the higher mean rate of return? Which sector has the higher median rate of return? (c) Determine the standard deviation rate of return for each sector. In finance, the standard deviation rate of return is called risk. Typically, an investor “pays” for a higher return by accepting more risk. Is the investor paying for higher returns for these sectors? Do you think the higher returns are worth the cost? Explain.

Temperatures It is well known that San Diego has milder weather than Chicago, but which city has more deviation from normal temperatures? Use the following data, which represent the deviation from normal high temperatures for a random sample of days. In which city would you rather be a meteorologist? Why?

The Empirical Rule The Stanford–Binet Intelligence Quotient (IQ) measures intelligence. IQ scores have a bell shaped distribution with a mean of 100 and a standard deviation of 15. (a) What percentage of people has an IQ score between 70 and 130? (b) What percentage of people has an IQ score less than 70 or greater than 130? (c) What percentage of people has an IQ score greater than 130?

The Empirical Rule SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114. Source: College Board (a) What percentage of SAT scores is between 401 and 629? (b) What percentage of SAT scores is less than 401 or greater than 629? (c) What percentage of SAT scores is greater than 743?

The Empirical Rule The weight, in grams, of the pair of kidneys in adult males between the ages of 40 and 49 has a bell shaped distribution with a mean of 325 grams and a standard deviation of 30 grams. (a) About 95% of kidney pairs will be between what weights? (b) What percentage of kidney pairs weighs between 235 grams and 415 grams? (c) What percentage of kidney pairs weighs less than 235 grams or more than 415 grams? (d) What percentage of kidney pairs weighs between 295 grams and 385 grams?

The Empirical Rule The distribution of the length of bolts has a bell shape with a mean of 4 inches and a standard deviation of 0.007 inch. (a) About 68% of bolts manufactured will be between what lengths? (b) What percentage of bolts will be between 3.986 inches and 4.014 inches? (c) If the company discards any bolts less than 3.986 inches or greater than 4.014 inches, what percentage of bolts manufactured will be discarded? (d) What percentage of bolts manufactured will be between 4.007 inches and 4.021 inches?

Which Professor? Suppose Professor Alpha and Professor Omega each teach Introductory Biology. You need to decide which professor to take the class from and have just completed your Introductory Statistics course. Records obtained from past students indicate that students in Professor Alpha’s class have a mean score of 80% with a standard deviation of 5%, while past students in Professor Omega’s class have a mean score of 80% with a standard deviation of 10%. Decide which instructor to take for Introductory Biology using a statistical argument.

Larry Summers Lawrence Summers (former Secretary of the Treasury and former president of Harvard) infamously claimed that women have a lower standard deviation IQ than men. He went on to suggest that this was a potential explanation as to why there are fewer women in top math and science positions. Suppose an IQ of 145 or higher is required to be a researcher at a top-notch research institution. Use the idea of standard deviation, the Empirical Rule, and the fact that the mean and standard deviation IQ of humans is 100 and 15, respectively, to explain Summers’ argument.

Chebyshev’s Inequality In December 2014, the average price of regular unleaded gasoline excluding taxes in the United States was $3.06 per gallon, according to the Energy Information Administration. Assume that the standard deviation price per gallon is $0.06 per gallon to answer the following. (a) What minimum percentage of gasoline stations had prices within 3 standard deviations of the mean? (b) What minimum percentage of gasoline stations had prices within 2.5 standard deviations of the mean? What are the gasoline prices that are within 2.5 standard deviations of the mean? (c) What is the minimum percentage of gasoline stations that had prices between $2.94 and $3.18?

Chebyshev’s Inequality According to the U.S. Census Bureau, the mean of the commute time to work for a resident of Boston, Massachusetts, is 27.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes to answer the following: (a) What minimum percentage of commuters in Boston has a commute time within 2 standard deviations of the mean? (b) What minimum percentage of commuters in Boston has a commute time within 1.5 standard deviations of the mean? What are the commute times within 1.5 standard deviations of the mean? (c) What is the minimum percentage of commuters who have commute times between 3 minutes and 51.6 minutes?

Comparing Standard Deviations The standard deviation of batting averages of all teams in the American League is 0.008. The standard deviation of all players in the American League is 0.02154. Why is there less variability in team batting averages?

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 range, population variance, and population standard deviation for the data. (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 range, population variance, and population standard deviation. 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 range, population variance, and population standard deviation. 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 range, population variance, and population standard deviation. How do they compare to the originals?

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 sample standard deviation. Then recompute the sample standard deviation assuming that the individual whose IQ is 106 is accidentally recorded as 160. For each sample size, state what happens to the standard deviation. Comment on the role that the number of observations plays in resistance.

Identical Values Compute the sample standard deviation of the following test scores: 78, 78, 78, 78. What can be said about a data set in which all the values are identical?

Buying a Car The following data represent the asking price, in dollars, for a random sample of 2010 coupes and a random sample of 2010 Chevy Camaros. (a) Find the mean and standard deviation price for each sample. (b) Explain why the mean is higher for Camaros yet the standard deviation is less.

Blocking and Variability Recall that blocking refers to the idea that we can reduce the variability in a variable by segmenting the data by some other variable. The given data represent the recumbent length (in centimeters) of a sample of 10 males and 10 females who are 40 months of age. (a) Determine the standard deviation of recumbent length for all 20 observations. (b) Determine the standard deviation of recumbent length for the males. (c) Determine the standard deviation of recumbent length for the females. (d) What effect does blocking by gender have on the standard deviation of recumbent length for each gender?

Mean Absolute Deviation Another measure of variation is the mean absolute deviation. It is computed using the formula MAD = Compute the mean absolute deviation of the data in Problem 11 and compare the results with the sample standard deviation.

Coefficient of Skewness Karl Pearson developed a measure that describes the skewness of a distribution, called the coefficient of skewness. The formula is Skewness = The value of this measure generally lies between -3 and +3. The closer the value lies to -3, the more the distribution is skewed left. The closer the value lies to +3, the more the distribution is skewed right. A value close to 0 indicates a symmetric distribution. Find the coefficient of skewness of the following distributions and comment on the skewness. (a) Mean = 50, median = 40, standard deviation = 10 (b) Mean = 100, median = 100, standard deviation = 15 (c) Mean = 400, median = 500, standard deviation = 120 (d) Compute the coefficient of skewness for the data in Problem 23. (e) Compute the coefficient of skewness for the data in Problem 24.

Diversification A popular theory in investment states that you should invest a certain amount of money in foreign investments to reduce your risk. The risk of a portfolio is defined as the standard deviation of the rate of return. Refer to the raph in the next column, which depicts the relation between risk (standard deviation of rate of return) and reward (mean rate of return). (a) Determine the average annual return and level of risk in a portfolio that is 10% foreign. (b) Determine the percentage that should be invested in foreign stocks to best minimize risk. (c) Why do you think risk initially decreases as the percentage of foreign investments increases? (d) A portfolio that is 30% foreign and 70% American has a mean rate of return of about 15.8%, with a standard deviation of 14.3%. According to Chebyshev’s Inequality, at least 75% of returns will be between what values? According to Chebyshev’s Inequality, at least 88.9% of returns will be between what two values? Should an investor be surprised if she has a negative rate of return? Why?

More Spread? The data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds. (a) Determine the mean and standard deviation of each data set. (b) Based only on the standard deviation, which data set has more spread? (c) What proportion of the observations is within one standard deviation of the mean for each data set? (d) The coefficient of variation, CV, is defined as the ratio of the standard deviation to the mean of a data set, so CV = The coefficient of variation is unitless and allows for comparison in spread between two data sets by describing the amount of spread per unit mean. After all, larger numbers will likely have a larger standard deviation simply due to the size of the numbers. Compute the coefficient of variation for both data sets. Which data set do you believe has more “spread”? (e) Let’s take this idea one step further. The following data represent the height of a random sample of 8 male college students. The data set on the left has their height measured in inches, and the data set on the right has their height measured in centimeters. For each data set, determine the mean and the standard deviation. Would you say that the height of the males is more dispersed based on the standard deviation of height measured in centimeters? Why? Now, determine the coefficient of variation for each data set. What did you find?

Sullivan Survey Choose two quantitative variables from the SullivanStats Survey I at www.pearsonhighered.com/sullivanstats for which a comparison is reasonable, such as number of hours of television versus number of hours of Internet. Draw a histogram for each variable. Which variable appears to have more dispersion? Determine the range and standard deviation of each variable. Based on the numerical measure, which variable has more dispersion?

Sullivan Survey Choose any quantitative variable from the SullivanStats Survey I at www.pearsonhighered.com/sullivanstats. Now choose a qualitative variable, such as gender or political philosophy. Determine the range and standard deviation by the qualitative variable chosen. For example, if you choose gender as the qualitative variable, determine the range and standard deviation by gender. Does there appear to be any difference in the measure of dispersion for each level of the qualitative variable?

Would it be appropriate to say that a distribution with a standard deviation of 10 centimeters is more dispersed than a distribution with a standard deviation of 5 inches? Support your position.

What is meant by the phrase degrees of freedom as it pertains to the computation of the sample standard deviation?

Are any of the measures of dispersion mentioned in this section resistant? Explain.

What does it mean when a statistic is biased?

What makes the range less desirable than the standard deviation as a measure of dispersion?

In one of Sullivan’s statistics sections, the standard deviation of the heights of all students was 3.9 inches. The standard deviation of the heights of males was 3.4 inches and the standard deviation of females was 3.3 inches. Why is the standard deviation of the entire class more than the standard deviation of the males and females considered separately?

Explain how standard deviation measures spread. Your explanation includes the computation of the same standard deviation for two data sets: Data set I: 3, 4, 5; Data set II: 0, 4, 8.

Which of the following would have a higher standard deviation? (a) IQ of students on your campus or (b) IQ of residents in your home town? Why?

Develop a sample of size n = 8 such that = 15 and s = 0.

Draw two histograms with different standard deviations and label them I and II. Which histogram has the larger standard deviation?

Fast Pass In 2000, the Walt Disney Company created the “fast pass” —a ticket issued to a rider for a popular ride and the rider is told to return at a specific time during the day. At that time, the rider is allowed to bypass the regular line, thereby reducing the wait time for that particular rider. When compared to wait times prior to creating the fast pass, overall wait times for rides at the park increased, on average, yet patrons to the park indicated they were happier with the fast-pass system. Use the concepts of central tendency and dispersion to explain why.