Solved: A Fish Story Ethan and Drew went on a 10-day

Problem 1AYU The sum of the deviations about the mean always equals_____.

Problem 2AYU 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.

Problem 3AYU 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.

Problem 4AYU 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.

Find the population variance and standard deviation or the sample variance and standard deviation as indicated. Sample: 20, 13, 4, 8, 10

Problem 6AYU 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

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

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

Problem 10AYU Find the population variance and standard deviation or the sample variance and standard deviation as indicated. Population: 4, 10, 12, 12, 13, 21

Crash Test Results The Insurance Institute for Highway Safety crashed the 2010 Ford Fusion four times at 5 miles per hour. The cost of repair for each of the four crashes are as follows: $2529, $1889, $2610, $1073 Compute the range, sample variance, and sample standard deviation cost of repair.

Problem 12AYU 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 range, sample variance, and sample standard deviation phone bill.

Problem 13AYU 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 up for 28 days. The concrete’s strength is then measured (in psi).The following data represent the strength of nine randomly selected casts: 3960, 4090, 3200, 3100, 2940, 3830, 4090, 4040, 3780 Compute the range and sample standard deviation for the strength of the concrete (in psi).

Problem 14AYU 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 range and sample standard deviation of flight time.

Problem 15AYU Which histogram depicts a higher standard deviation? Justify your answer. (a) (b)

Match the histograms on the following page to the summary statistics given.

Problem 17AYU 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. Tap 7.64 7.45 7.45 7.10 7.47 7.56 7.50 7.47 7.68 7.69 7.52 7.47 Bottled 5.18 5.28 5.09 5.26 5.26 5.13 5.20 5.26 5.02 5.23 5.21 5.24 (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?

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 results for six study participants are listed in the table. 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) Which color has more dispersion using the range as the measure of dispersion? (b) Which color has more dispersion using the standard deviation as the measure of dispersion?

Pulse Rates The following data 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: Ethan 9 24 8 9 5 8 9 10 8 10 Drew 15 2 3 18 20 1 17 2 19 3 (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) 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. (c) Discuss limitations of the range as a measure of dispersion.

Soybean Yield The following data 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 27, 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. (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 28, 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. (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.

Problem 25AYU Problem Which Car Would You Buy? Suppose that you are in the market to purchase a car. With gas prices on the rise, 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 gas. You conduct this experiment 15 times on each car and record the number of miles driven. CAR 1 228 223 178 220 220 233 233 271 219 223 217 214 189 236 248 CAR 2 277 164 326 215 259 217 321 263 160 257 239 230 183 217 230 Describe each data set. That is, determine the shape, center, and spread. Which car would you buy and why?

Problem 26AYU Problem Which Investment Is Better? You have received a yearend 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). MUTUAL FUND A 1.3 0.3 0.6 6.8 5.0 5.2 4.8 2.4 3.0 1.8 7.3 8.6 3.4 3.8 ?1.3 6.4 1.9 ?0.5 ?2.3 3.1 MUTUAL FUND B ?5.4 6.7 11.9 4.3 4.3 3.5 10.5 2.9 3.8 5.9 ?6.7 1.4 8.9 0.3 ?2.4 ?4.7 ?1.1 3.4 7.7 12.9 Describe each data set. That is, determine the shape, center, and spread. Which mutual fund would you invest in and why?

Rates of Return of Stocks Stocks may be categorized by industry. The data on the following page represent the 5?year rates of return (in percent) for a sample of consumer goods stocks and energy stocks ending November 10, 2010. (a) Determine the mean and the median rate of return for each industry. Which sector has the higher mean rate of return? Which sector has the higher median rate of return? (b) Determine the standard deviation for each industry. 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 in this instance? Do you think the higher returns are worth the cost?

Temperatures It is well known that San Diego has milder weather than Chicago, but which city has more deviation from normal temperatures over the course of a month? Use the following data, which represent the deviation from normal high temperatures for each day in October 2010. In which city would you rather be a meteorologist? Why?

Problem 29AYU Problem The Empirical Rule One measure of intelligence is the Stanford–Binet Intelligence Quotient (IQ). 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?

Problem 30AYU Problem 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, 2010 (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?

Problem 31AYU Problem 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?

Problem 32AYU Problem 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 2010, 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?

Problem 37AYU Problem 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?

Problem 39AYU Problem 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. SAMPLE OF SIZE 5 106 92 98 103 100 SAMPLE OF SIZE 12 106 92 98 103 98 124 83 70 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

Blocking and Variability 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 \(\mathrm{MAD}=\frac{\Sigma|x_i-\bar x|}{n}\) 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 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 1. (e) Compute the coefficient of skewness for the data in Problem 2. Problem 1 The Empirical Rule The following data represent the weights (in grams) of a random sample of 50 M&M plain candies. Problem 2 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.

Problem 44AYU Problem 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 graph on the following page, which depicts the relation between risk (standard deviation of rate of return) and reward (mean rate of return). How Foreign Stocks Benefit a Domestic Portfolio (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 percent 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?

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 \(\mathrm{CV}=\frac{\text { standard deviation }}{\text { mean }}\) 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 any two quantitative variables from the Sullivan Survey 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 Sullivan Survey. 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 chose 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.

Problem 49AYU Problem 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?

Problem 52AYU Problem 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. In your explanation include the computation of the same standard deviation for two data sets: Data set 1: 3, 4, 5; Data set 2: 0,4 ,8.

Problem 55AYU Problem 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 \(\bar x=15\) 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 fast-pass ticket is issued to a rider for the popular rides and the rider is told to return at a specific time during the day. At that time, the rider is allowed to by-pass 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.