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Introduction To The Practice Of Statistics 6th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions

Babe Ruth and Mark McGwire. Babe Ruth hit 60 home runs in 1927, a record that stood until Mark McGwire hit 70 in 1998. A proper comparison of Ruth and McGwire should include their historical context. Here are the number of home runs by the major league leader for each year in baseball history, 1876
By-products from DDT. By-products from the pesticide DDT were major threats to the survival of birds of prey until use of DDT was banned at the end of 1972. Can time plots show the effect of the ban? Here are two sets of data for bald eagles nesting in the forests of northwestern Ontario.43 The
Two distributions. If two distributions have exactly the same mean and standard deviation, must their histograms have the same shape? If they have the same five-number summary, must their histograms have the same shape? Explain.
Spam filters. A university department installed a spam filter on its computer system. During a 21-day period, 6693 messages were tagged as spam. How much spam you get depends on what your online habits are. Here are the counts for some students and faculty in this department(with log-in IDs
Household size and household income. Rich and poor households differ in ways that go beyond income. Figure 1.44 displays histograms that compare the distributions of household size (number of people) for low-income and high-income households in 2002.42 Low-income households had incomes less than
What graph would you use? What type of graph or graphs would you plan to make in a study of each of the following issues?(a) What makes of cars do students drive? How old are their cars?(b) How many hours per week do students study?How does the number of study hours change during a semester?(c)
Weights are not Normal. The heights of people of the same sex and similar ages follow Normal distributions reasonably closely. Weights, on the other hand, are not Normally distributed. The weights of women aged 20 to 29 have mean 141.7 pounds and median 133.2 pounds. The first and third quartiles
Internet service. Late in 2003, there were 77.4 million residential subscribers to Internet service in the United States. The numbers of subscribers claimed by the top 10 providers of service were as follows.41 (There is some doubt about the accuracy of these claims.)Service Subscribers Service
Distance-learning courses. The 222 students enrolled in distance-learning courses offered by a college ranged from 18 to 64 years of age. The mode of their ages was 19. The median age was 31.40 Explain how this can happen.
Product preference. Product preference depends in part on the age, income, and gender of the consumer. A market researcher selects a large sample of potential car buyers. For each consumer, she records gender, age, household income, and automobile preference. Which of these variables are
Biological clocks. Many plants and animals have “biological clocks” that coordinate activities with the time of day. When researchers looked at the length of the biological cycle in the plant Arabidopsis by measuring leaf movements, they found that the length of the cycle is not always 24
Leisure time for college students. You want to measure the amount of “leisure time” that college students enjoy. Write a brief discussion of two issues:(a) How will you define “leisure time”?(b) How will you measure leisure time?
Compare two Normal curves. In Exercise 1.99, we worked with the distribution of ISTEP scores on the English/language arts portion of the exam for tenth-graders. We used the fact that the distribution of scores for the 76,531 students who took the exam was approximately N(572, 51).These students
Park space and population. Below are data on park and open space in several U.S. cities with high population density.38 In this table, population is reported in thousands of people, and park and open space is called open space, with units of acres.City Population Open space Baltimore 651 5,091
Use software to generate more data. Use software to generate 100 observations from the uniform distribution described in Exercise 1.108.Make a histogram of these observations. How does the histogram compare with the density curve in Figure 1.37? Make a Normal quantile plot of your data. According
Use software to generate some data. Use software to generate 100 observations from the standard Normal distribution. Make a histogram of these observations. How does the shape of the histogram compare with a Normal density curve? Make a Normal quantile plot of the data.Does the plot suggest any
Logging in Borneo. The study of the effects of logging on tree counts in the Borneo rain forest(Exercise 1.80, page 51) used statistical methods that are based on Normal distributions. Make Normal quantile plots for each of the three groups of forest plots. Are the three distributions roughly
Three varieties of flowers. The study of tropical flowers and their hummingbird pollinators(Exercise 1.78, page 51) measured lengths for three varieties of Heliconia flowers. We expect that such biological measurements will have roughly Normal distributions.(a) Make Normal quantile plots for each
Density of the earth. We expect repeated careful measurements of the same quantity to be approximately Normal. Make a Normal quantile plot for Cavendish’s measurements in Exercise 1.40(page 28). Are the data approximately Normal? If not, describe any clear deviations from Normality.
CHALLENGE Four Normal quantile plots. Figure 1.42 shows four Normal quantile plots for data that you have seen before, without scales for the variables plotted. In scrambled order, they are:1. The IQ scores in the histogram of Figure 1.7(page 14).2. The tuition and fee charges of Massachusetts
Electrical meters. The distance between two mounting holes is important to the performance of an electrical meter. The manufacturer measures this distance regularly for quality control purposes, recording the data as thousandths of an inch more than 0.600 inches. For example, 0.644 is recorded as
Carbon dioxide emissions. Figure 1.40 is a Normal quantile plot of the emissions of carbon dioxide (CO2) per person in 48 countries, from Table 1.6 (page 26). In what way is this distribution non-Normal? Comparing the plot with Table 1.6, which countries would you call outliers?
Heart rates of runners. Figure 1.39 is a Normal quantile plot of the heart rates of the 200 male runners in the study described in Exercise 1.81(page 51). The distribution is close to Normal. How can you see this? Describe the nature of the small deviations from Normality that are visible in the
CHALL ENGE Outliers for Normal distributions.Continue your work from the previous two exercises. The percent of the observations that are suspected outliers according to the 1.5 × IQR rule is the same for any Normal distribution. What is this percent?
CH ALLENGE IQR for Normal distributions. Continue your work from the previous exercise. The interquartile range IQR is the distance between the first and third quartiles of a distribution.(a) What is the value of the IQR for the standard Normal distribution?(b) There is a constant c such that IQR =
CH ALLENGE Quartiles for Normal distributions. The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75.(a) What are the quartiles of the standard Normal distribution?(b) Using your numerical values from (a), write an equation that gives the quartiles of the N(μ,
Length of pregnancies. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days.(a) What percent of pregnancies last less than 240 days (that’s about 8 months)?(b) What percent of
Diagnosing osteoporosis. Osteoporosis is a condition in which the bones become brittle due to loss of minerals. To diagnose osteoporosis, an elaborate apparatus measures bone mineral density(BMD). BMD is usually reported in standardized form. The standardization is based on a population of healthy
Proportion of men with high cholesterol.Middle-aged men are more susceptible to high cholesterol than the young women of the previous exercise. The blood cholesterol levels of men aged 55 to 64 are approximately Normal with mean 222 mg/dl and standard deviation 37 mg/dl. What percent of these men
Proportion of women with high cholesterol. Too much cholesterol in the blood increases the risk of heart disease. Young women are generally less afflicted with high cholesterol than other groups.The cholesterol levels for women aged 20 to 34 follow an approximately Normal distribution with mean 185
Find the SAT quintiles. The quintiles of any distribution are the values with cumulative proportions 0.20, 0.40, 0.60, and 0.80. What are the quintiles of the distribution of SAT scores?
Find the ACT quartiles. The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75. What are the quartiles of the distribution of ACT scores?
How low is the bottom 20%? What SAT scores make up the bottom 20% of all scores?
How high is the top 10%? What SAT scores make up the top 10% of all scores?
Find the ACT percentile. Reports on a student’s ACT or SAT usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as a percent: the percent of all scores that were lower than this one. Jacob scores 17 on the ACT. What is his percentile?
Find the SAT percentile. Reports on a student’s ACT or SAT usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as a percent: the percent of all scores that were lower than this one. Tonya scores 1320 on the SAT. What is her percentile?
Find the SAT equivalent. Maria scores 29 on the ACT. Assuming that both tests measure the same thing, what score on the SAT is equivalent to Maria’s ACT score?
Find the ACT equivalent. Jose scores 1380 on the SAT. Assuming that both tests measure the same thing, what score on the ACT is equivalent to Jose’s SAT score?
Make another comparison. Jacob scores 17 on the ACT. Emily scores 680 on the SAT. Assuming that both tests measure the same thing, who has the higher score? Report the z-scores for both students.
Compare an SAT score with an ACT score.Tonya scores 1320 on the SAT. Jermaine scores 28 on the ACT. Assuming that both tests measure the same thing, who has the higher score? Report the z-scores for both students.
High IQ scores. The Wechsler Adult Intelligence Scale (WAIS) is the most common “IQ test.” The scale of scores is set separately for each age group and is approximately Normal with mean 100 and standard deviation 15. The organization MENSA, which calls itself “the high IQ society,” requires
Find some values of z. The Wechsler Adult Intelligence Scale (WAIS) is the most common“IQ test.” The scale of scores is set separately for each age group and is approximately Normal with mean 100 and standard deviation 15. People with WAIS scores below 70 are considered mentally retarded when,
Find more values of z. The variable Z has a standard Normal distribution.(a) Find the number z that has cumulative proportion 0.85.(b) Find the number z such that the event Z > z has proportion 0.40.
Find some values of z. Find the value z of a standard Normal variable Z that satisfies each of the following conditions. (If you use Table A, report the value of z that comes closest to satisfying the condition.) In each case, sketch a standard Normal curve with your value of z marked on the
Find more proportions. Using either Table A or your calculator or software, find the proportion of observations from a standard Normal distribution for each of the following events. In each case, sketch a standard Normal curve and shade the area representing the proportion.(a) Z ≤ −1.9(b) Z ≥
Find some proportions. Using either Table A or your calculator or software, find the proportion of observations from a standard Normal distribution that satisfies each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to
Acidity of rainwater. The Normal quantile plot in Figure 1.36 (page 71) shows that the acidity(pH) measurements for rainwater samples in Exercise 1.36 are approximately Normal. How well do these scores satisfy the 68–95–99.7 rule?To find out, calculate the mean ¯x and standard deviation s of
APPLET Use the Normal Curve applet. Use the Normal Curve applet for the standard Normal distribution to say how many standard deviations above and below the mean the quartiles of any Normal distribution lie.
Heights of women. The heights of women aged 20 to 29 are approximately Normal with mean 64 inches and standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches. What are the zscores for a woman 6 feet tall and a man 6 feet tall? What information
Binge drinking survey. One reason that Normal distributions are important is that they describe how the results of an opinion poll would vary if the poll were repeated many times. About 20%of college students say they are frequent binge drinkers. Think about taking many randomly chosen samples of
Horse pregnancies are longer. Bigger animals tend to carry their young longer before birth.The length of horse pregnancies from conception to birth varies according to a roughly Normal distribution with mean 336 days and standard deviation 3 days. Use the 68–95–99.7 rule to answer the following
Pregnancies and the 68–95–99.7 rule. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Use the 68–95–99.7 rule to answer the following questions.(a) Between what values
APPLET Use the Normal Curve applet. The 68–95–99.7 rule for Normal distributions is a useful approximation. You can use the Normal Curve applet on the text CD and Web site to see how accurate the rule is. Drag one flag across the other so that the applet shows the area under the curve between
Length of pregnancies. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Draw a density curve for this distribution on which the mean and standard deviation are correctly
Three density curves. Figure 1.38 displays three density curves, each with three points marked on it. At which of these points on each curve do the mean and the median fall?
Find the mean, the median, and the quartiles.What are the mean and the median of the uniform distribution in Figure 1.37? What are the quartiles?
Use a different range for the uniform distribution. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the outcomes are to be distributed uniformly between 0 and 4. Then the density curve of the outcomes has constant
A uniform distribution. If you ask a computer to generate “random numbers” between 0 and 1, you will get observations from a uniform distribution.Figure 1.37 graphs the density curve for a uniform distribution. Use areas under this density curve to answer the following questions.0 1 FIGURE 1.37
Sketch some density curves. Sketch density curves that might describe distributions with the following shapes:(a) Symmetric, but with two peaks (that is, two strong clusters of observations).(b) Single peak and skewed to the right.
Find the score that 60% of students will exceed. Consider the ISTEP scores, which are approximately Normal, N(572, 51). Sixty percent of the students will score above x on this exam. Find x.
What score is needed to be in the top 5%? Consider the ISTEP scores, which are approximately Normal, N(572, 51). How high a score is needed to be in the top 5% of students who take this exam?
Find another proportion. Consider the ISTEP scores, which are approximately Normal, N(572, 51). Find the proportion of students who have scores between 600 and 650. Use pictures of Normal curves similar to the ones given in Example 1.28 to illustrate your calculations.
Find the proportion. Consider the ISTEP scores, which are approximately Normal, N(572, 51). Find the proportion of students who have scores less than 600. Find the proportion of students who have scores greater than or equal to 600. Sketch the relationship between these two calculations using
Find another z-score. Consider the ISTEP scores, which we can assume are approximately Normal, N(572, 51). Give the z-score for a student who received a score of 500. Explain why your answer is negative even though all of the test scores are positive.
Find the z-score. Consider the ISTEP scores (see Exercise 1.99), which we can assume are approximately Normal, N(572, 51). Give the z-score for a student who received a score of 600.
Use the 68–95–99.7 rule. Refer to the previous exercise. Use the 68–95–99.7 rule to give a range of scores that includes 99.7% of these students.
Test scores. Many states have programs for assessing the skills of students in various grades. The Indiana Statewide Testing for Educational Progress (ISTEP) is one such program.35 In a recent year 76,531 tenth-grade Indiana students took the English/language arts exam. The mean score was 572 and
CHALLENGE Changing units from centimeters to inches. Refer to Exercise 1.56. Change the measurements from centimeters to inches by multiplying each value by 0.39. Answer the questions from the previous exercise and explain the effect of the transformation on these data.
A different type of mean. The trimmed mean is a measure of center that is more resistant than the mean but uses more of the available information than the median. To compute the 10% trimmed mean, discard the highest 10% and the lowest 10%of the observations and compute the mean of the remaining
CHALLENGE Changing units from inches to centimeters. Changing the unit of length from inches to centimeters multiplies each length by 2.54 because there are 2.54 centimeters in an inch. This change of units multiplies our usual measures of spread by 2.54. This is true of IQR and the standard
Guinea pigs. Find the quintiles (the 20th, 40th, 60th, and 80th percentiles) of the guinea pig survival times in Table 1.8 (page 29). For quite large sets of data, the quintiles or the deciles (10th, 20th, 30th, etc. percentiles) give a more detailed summary than the quartiles.
CH ALLENGE The density of the earth. Henry Cavendish(see Exercise 1.40, page 28) used ¯x to summarize his 29 measurements of the density of the earth.(a) Find x and s for his data.(b) Cavendish recorded the density of the earth as a multiple of the density of water. The density of water is almost
Compare three varieties of flowers. Exercise 1.78 reports data on the lengths in millimeters of flowers of three varieties of Heliconia. In Exercise 1.79 you found the mean and standard deviation for each variety. Starting from the x- and s-values in millimeters, find the means and standard
Weight gain. A study of diet and weight gain deliberately overfed 16 volunteers for eight weeks.The mean increase in fat was ¯x = 2.39 kilograms and the standard deviation was s = 1.14 kilograms.What are ¯x and s in pounds? (A kilogram is 2.2 pounds.)
Guinea pigs. Table 1.8 (page 29) gives the survival times of 72 guinea pigs in a medical study. Survival times—whether of cancer patients after treatment or of car batteries in everyday use—are almost always right-skewed. Make a graph to verify that this is true of these survival times. Then
Does your software give incorrect answers? This exercise requires a calculator with a standard deviation button or statistical software on a computer. The observations 20,001 20,002 20,003 have mean x = 20,002 and standard deviation s = 1.Adding a 0 in the center of each number, the next set
CHALLENGE A standard deviation contest. This is a standard deviation contest. You must choose four numbers from the whole numbers 0 to 20, with repeats allowed.(a) Choose four numbers that have the smallest possible standard deviation.(b) Choose four numbers that have the largest possible standard
CHALLENGE Deviations from the mean sum to zero.Use the definition of the mean ¯x to show that the sum of the deviations xi − ¯x of the observations from their mean is always zero. This is one reason why the variance and standard deviation use squared deviations.
Create another data set. Give an example of a small set of data for which the mean is larger than the third quartile.
CHALLENGE Create a data set. Create a set of 5 positive numbers (repeats allowed) that have median 10 and mean 7. What thought process did you use to create your numbers?
CHALLENGE Shakespeare’s plays. Look at the histogram of lengths of words in Shakespeare’s plays, Figure 1.15 (page 25). The heights of the bars tell us what percent of words have each length. What is the median length of words used by Shakespeare?Similarly, what are the quartiles? Give the
How does income change with education? Write a brief description of how the distribution of income changes with the highest level of education reached.Be sure to discuss center, spread, and skewness.Give some specifics read from the graph to back up your statements.
Find the 5th and 95th percentiles. About what are the positions of the 5th and 95th percentiles in the ordered list of incomes of the 14,959 people with a bachelor’s degree? Incomes outside this range do not appear in the boxplot. About what are the numerical values of the 5th and 95th
Income for people with bachelor’s degrees. The data include 14,959 people whose highest level of education is a bachelor’s degree.(a) What is the position of the median in the ordered list of incomes (1 to 14,959)? From the boxplot, about what is the median income of people with a bachelor’s
CHALLENGE Running and heart rate. How does regular running affect heart rate? The RUNNERS data set, described in detail in the Data Appendix, contains heart rates for four groups of people:Sedentary females Sedentary males Female runners (at least 15 miles per week)Male runners (at least 15 miles
CHALLENGE Effects of logging in Borneo.“Conservationists have despaired over destruction of tropical rainforest by logging, clearing, and burning.” These words begin a report on a statistical study of the effects of logging in Borneo. Researchers compared forest plots that had never been logged
Compare the three varieties of flowers. The biologists who collected the flower length data in the previous exercise compared the three Heliconia varieties using statistical methods based on ¯x and s.(a) Find ¯x and s for each variety.(b) Make a stemplot of each set of flower lengths.Do the
Hummingbirds and flowers. Different varieties of the tropical flower Heliconia are fertilized by different species of hummingbirds. Over time, the lengths of the flowers and the form of the hummingbirds’ beaks have evolved to match each other. Here are data on the lengths in millimeters of three
APPLET Mean and median for five observations.Place five observations on the line in the Mean and Median applet by clicking below it.(a) Add one additional observation without changing the median. Where is your new point?(b) Use the applet to convince yourself that when you add yet another
APPLET Mean and median for three observations.In the Mean and Median applet, place three observations on the line by clicking below it, two close together near the center of the line and one somewhat to the right of these two.(a) Pull the single rightmost observation out to the right. (Place the
APPLET Mean and median for two observations.The Mean and Median applet allows you to place observations on a line and see their mean and median visually. Place two observations on the line, by clicking below it. Why does only one arrow appear?
IQ scores. Many standard statistical methods that you will study in Part II of this book are intended for use with distributions that are symmetric and have no outliers. These methods start with the mean and standard deviation, ¯x and s. For example, standard methods would typically be used for
The density of the earth. Many standard statistical methods that you will study in Part II of this book are intended for use with distributions that are symmetric and have no outliers. These methods start with the mean and standard deviation, ¯x and s. Two examples of scientific data for which
Distributions for time spent studying. Exercise 1.41 (page 28) presented data on the nightly study time claimed by first-year college men and women.The most common methods for formal comparison of two groups use ¯x and s to summarize the data.We wonder if this is appropriate here. Look at your
CHALLENGE Hurricanes and losses. A discussion of extreme weather says: “In most states, hurricanes occur infrequently. Yet, when a hurricane hits, the losses can be catastrophic. Average annual losses are not a meaningful measure of damage from rare but potentially catastrophic events.”31 Why
Metabolic rates. Calculate the mean and standard deviation of the metabolic rates in Example 1.19(page 41), showing each step in detail. First find the mean ¯x by summing the 7 observations and dividing by 7. Then find each of the deviations xi − x and their squares. Check that the deviations
How does the median change? The firm in Exercise 1.67 gives no raises to the clerks and junior accountants, while the owner’s take increases to$455,000. How does this change affect the mean?How does it affect the median?
Be careful about how you treat the zeros. In computing the median income of any group, some federal agencies omit all members of the group who had no income. Give an example to show that the reported median income of a group can go down even though the group becomes economically better off. Is this
Mean versus median. A small accounting firm pays each of its five clerks $35,000, two junior accountants $80,000 each, and the firm’s owner$320,000. What is the mean salary paid at this firm?How many of the employees earn less than the mean? What is the median salary?
Mean versus median for oil wells. Exercise 1.39(page 28) gives data on the total oil recovered from 64 wells. Your graph in that exercise shows that the distribution is clearly right-skewed.(a) Find the mean and median of the amounts recovered. Explain how the relationship between the mean and the

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