Exploring Data

1. Use the back-to-back stemplot in Figure $1.15$to write a few sentences comparing the number of pairs of shoes owned by males and females. Be sure to address shape, center, spread, and outliers.

The low outlier is Alaska. What percent of Alaska residents are $65$ older?

(a) $0.68$

(b) $6.8$

(c) $8.8$

(d) $16.8$

(e) $68$

Use the back-to-back stem plot in Figure $1.15$ to write a few sentences comparing the number of pairs of shoes owned by males and females. Be sure to address shape, center, spread, and outliers. 

Ignoring the outlier, the shape of the distribution is,

(a) skewed to the right and (c) skewed to the left. (e) skewed to the middle.

(b) roughly symmetric (d) bimodal.

Use the back-to-back stem plot in Figure $1.15$ to write a few sentences comparing the number of pairs of shoes owned by males and females. Be sure to address shape, center, spread, and outliers. 

The center of the distribution is close to-

$\left(a\right) 13.3\%. \left(b\right) 12.8\%. \left(c\right) 12.0\%. \left(d\right) 11.6\%. \left(e\right) 6.8\% to 16.8\%$

Many people believe that the distribution of IQ scores follows a “bell curve,” like the one shown in the margin. But is this really how such scores are distributed? The IQ scores of 60 fifth-grade students chosen at random from one school are shown below. 

Describe what you see. Is the distribution bell-shaped?

Many people believe that the distribution of IQ scores follows a “bell curve,” like the one shown in the margin. But is this really how such scores are distributed? The IQ scores of 60 fifth-grade students chosen at random from one school are shown below.

Construct a histogram that displays the distribution of IQ scores effectively

Draw a correct histogram to replace Billy’s graph of the word length data from Caution 2. 

Draw a more meaningful graph of the first-name length data from Caution $4$

Is this a bar graph or a histogram? 

About 1.6 million first-year students enroll in colleges and universities each year. What do they plan to study? The graph displays data on the per cent of first-year students who plan to major in several discipline areas.

Is this a bar graph or a histogram? Explain.

About 1.6 million first-year students enroll in colleges and universities each year. What do they plan to study? The graph displays data on the percent of first-year students who plan to major in several discipline areas.

Would it be correct to describe this distribution as right-skewed? Why or why not?

Feeling sleepy? Students in a college statistics class responded to a survey designed by their teacher. One of the survey questions was “How much sleep did you get last night?” Here are the data (in hours): 

(a) Make a dot plot to display the data.

(b) Describe the overall pattern of the distribution

and any deviations from that pattern.

Olympic gold! The following table displays the total number of gold medals won by a sample of countries in the $2008$ Summer Olympic Games in China.

(a) Make a dot plot to display these data. Describe the overall pattern of the distribution and any deviations from that pattern.

(b) Overall, $204$ countries participated in the $2008$ Summer Olympics, of which $55$ won at least one gold medal. Do you believe that the sample of countries listed in the table is representative of this larger population? Why or why not? 

U.S. women’s soccer—$2004$ Earlier, we examined data on the number of goals scored by the U.S. women’s soccer team in games during the $2004$ season. The dot plot below displays the goal differential for those same games, computed as the U.S. score minus the opponent’s score

(a) Explain what the two dots above $-2$ represent.

(b) What does the graph tell us about how well the team did in $2004$? Be specific.

Fuel efficiency In an earlier example, we examined data on highway gas mileages of the model year $2009$ midsize cars. The dot plot below shows the difference (highway – city) in EPA mileage ratings for each of the $24$car models from the earlier example. 

(a) Explain what the dot above $6$represents.

(b) What does the graph tell us about fuel economy in the city versus on the highway for these car models? Be specific. 

Dates on coins

(a) Sketch a dot-plot for a distribution that is skewed to the left.

(b) Suppose that you and your friends emptied your pockets of coins and recorded the year marked on each coin. The distribution of dates would be skewed to the left. Explain why.

Phone numbers the dot-plot below displays the last digit of $100$ phone numbers chosen at random from a phone book. Describe the shape of the distribution. Does this shape make sense to you? Explain.

The chapter-opening Case Study described research by Teresa Amabile investigating whether external rewards would promote creativity in children’s artwork. Dr. Amabile conducted another study involving college students, who were divided into two groups using a chance process (like drawing names from a hat). The students in one group were given a list of statements about external reasons

(E) for writing, such as public recognition, making money, or pleasing their parents. Students in the other group were given a list of statements about internal reasons (I) for writing, such as expressing yourself and

enjoying playing with words. Both groups were then instructed to write a poem about laughter.

 Each student’s poem was rated separately by $12$ different poets using a creativity scale. The $12$ poets’ ratings of each student’s poem were averaged to obtain an overall creativity score.

A dot-plot of the two groups’ creativity scores is shown below. Compare the two distributions. What do you conclude about whether external rewards promote creativity?

Healthy cereal? Researchers collected data on $77$ brands of cereal at a local supermarket.$25$ For each brand, the sugar content (grams per serving) and the shelf in the store on which the cereal was located ($1$ = bottom, $2$ = middle, $3$ = top) were recorded. A dot-plot of the data is shown below. Compare the three distributions. Critics claim that supermarkets tend to put sugary kids’ cereals on lower shelves, where the kids can see them. Do the data from this study support this claim? 

Where do the young live? Below is a stem plot of the percent of residents aged $25$ to $34$ in each of the $50$ states. As in the stem plot for older residents (page 35), the stems are whole percent and the leaves are tenths of a percent. This time, each stem has been split in two, with values having leaves $0$ through$4$ placed on one stem, and values ending in $5$ through $9$ placed on another stem.

(a) Why did we split stems?

(b) Utah has the highest percentage of residents aged $25$ to $34$. What is that percent? Why do you think Utah has an unusually high percentage of residents in this age group?

(c) Describe the shape, center, and spread of the distribution, ignoring Utah.

Watch that caffeine! The U.S. Food and Drug Administration (USFDA) limits the amount of caffeine in a $12$-ounce can of carbonated beverage to$72$ milligrams. That translates to a maximum of$48$ milligrams of caffeine per $8$-ounce serving. Data on the caffeine content of popular soft drinks (in milligrams per $8$-ounce serving) are displayed in the stem plot below.

(a) Why did we split stems?

(b) Give an appropriate key for this graph.

(c) Describe the shape, center, and spread of the distribution. Compare the caffeine content of these drinks with the USFDA’s limit.

El Niño and the monsoon It appears that El Niño, the periodic warming of the Pacific Ocean west of South America, affects the monsoon rains that are essential for agriculture in India. Here are the monsoon rains (in millimeters) for the $23$ strong El Niño years between $1871$ and $2004:: 26$ 

(a) To make a stem plot of these rainfall amounts, round the data to the nearest 10, so that stems are hundreds of millimeters and leaves are tens of millimeters. Make two stem plots, with and without splitting

the stems. Which plot do you prefer?

(b) Describe the shape, center, and spread of the distribution.

(c) The average monsoon rainfall for all years from $1871$ to $2004$ is about$850$ millimeters. What effect does El Niño appear to have on monsoon rains?

Shopping spree A marketing consultant observed 50 consecutive shoppers at a supermarket. One variable of interest was how much each shopper spent in the store. Here are the data (in dollars), arranged in

increasing order: 

(a) Round each amount to the nearest dollar. Then make a stem $50$plot using tens of dollars as the stems and dollars as the leaves.

(b) Make another stem plot of the data by splitting stems. Which of the plots shows the shape of the distribution better?

(c) Write a few sentences describing the amount of money spent by shoppers at this supermarket.

Do women study more than men? We asked the students in a large first-year college class how many minutes they studied on a typical weeknight. Here are

the responses of random samples of $30$ women and $30$ men from the class:

(a) Examine the data. Why are you not surprised that most responses are multiples of 10 minutes? Are there any responses you consider suspicious?

(b) Make a back-to-back stem plot to compare the two samples. Does it appear that women study more than men (or at least claim that they do)? Justify your answer.

Basketball playoffs Here are the scores of the games played in the California Division I-AAA high school basketball playoffs 

On the same day, the final scores of games in the Division

V-AA were

(a) Construct a back-to-back stem plot to compare the points scored by the $32$ teams in the Division I-AAA playoffs and the $24$ teams in the Division V-AA

playoffs.

(b) Write a few sentences comparing the two distributions.

Returns on common stocks The return on a stock is the change in its market price plus any dividend payments made. The total return is usually expressed as a percent of the beginning price. The figure below shows a histogram of the distribution of the monthly returns for all common stocks listed on U.S. markets from January 1985 to September 2007 (273 months).28 The extreme low outlier represents the market crash of October 1987, when stocks lost 23% of their value in one month.

(a) Ignoring the outliers, describe the overall shape of the distribution of monthly returns.

(b) What is the approximate center of this distribution?

(c) Approximately what were the smallest and largest monthly returns, leaving out the outliers?

(d) A return less than zero means that stocks lost value in that month. About what percent of all months had returns less than zero?

Shakespeare The histogram below shows the distribution of lengths of words used in Shakespeare’s plays.$29$ Describe the shape, center, and spread of this distribution. 

Traveling to work How long do people travel each day to get to work? The following table gives the average travel times to work (in minutes) for workers in each state and the District of Columbia who is at least 16 years old and don’t work at home.

(a) Make a histogram of the travel times using classes of width $2$ minutes, starting at $14$ minutes. That is, the first class is $14$ to $16$ minutes, the second is $16$ to $18$ minutes, and so on.

(b) The shape of the distribution is a bit irregular. Is it closer to symmetric or skewed? About where is the center of the data? What is the spread in terms of the smallest and largest values? Are there any outliers?

Carbon dioxide emissions Burning fuels in power plants and motor vehicles emit carbon dioxide

$\left({CO}_2\right)$, which contributes to global warming. The table below displays ${CO}_2$ emissions per person from countries with populations of at least $20$ million.

(a) Make a histogram of the data using classes of width $2$, starting at $0$

(b) Describe the shape, center, and spread of the distribution. Which countries are outliers?

DRP test scores There are many ways to measure the reading ability of children. One frequently used test is the Degree of Reading Power (DRP). In a research study on third-grade students, the DRP was administered to $44$ students $32$ Their scores were: Make a histogram to display the data. Write a paragraph describing the distribution of DRP scores. 

Drive time Professor Moore, who lives a few miles outside a college town, records the time he takes to drive to the college each morning. Here are the times (in minutes) for $42$ consecutive weekdays: 

Make a histogram of these drive times. Is the distribution roughly symmetric, clearly skewed, or neither? Are there any clear outliers?

The statistics of writing style Numerical data can distinguish different types of writing, and sometimes even individual authors. Here are data on the percent of words of $1$ to $15$ letters used in articles in Popular science magazine: 

(a) Make a histogram of this distribution. Describe its shape, center, and spread.

(b) How does the distribution of lengths of words used in Popular science compare with the similar distribution for Shakespeare’s plays in Exercise $52$? Look in particular at short words ($2, 3,$ and $4$ letters) and very long words (more than $10$ letters).

Chest out, Soldier! In$1846$, a published paper provided chest measurements (in inches) of $5738$

Scottish militiamen. The table below summarizes the data

(a) Make a histogram.

(b) Describe the shape, center, and spread of the chest measurements distribution. Why might this

information be useful?

Paying for championships Does pay high salaries to lead to more victories in professional sports? The New York Yankees have long been known for having Major League Baseball’s highest team payroll. And over the years, the team has won many championships. This strategy didn’t pay off in $2008$ when the Philadelphia Phillies won the World Series. Maybe the Yankees didn’t spend enough money that year. The graph below shows histograms of the salary distributions for the two teams during the $2008$ season. Why can’t you use this graph to effectively compare the team payrolls? 

Paying for championships Refer to Exercise $59$. Here is another graph of the $2008$ salary distributions for the Yankees and the Phillies. Write a few sentences comparing these two distributions. 

Birth months Imagine asking a random sample of $60$ students from your school about their birth months. Draw a plausible graph of the distribution of birth months. (Hint: Should you use a bar graph or a histogram?)

Die rolls Imagine rolling a fair, six-sided die $60$ times. Draw a plausible graph of the distribution of die rolls. (Hint: Should you use a bar graph or a histogram?)

Who makes more? A manufacturing company is reviewing the salaries of its full-time employees below the executive level at a large plant. The clerical staff is almost entirely female, while a majority of the production workers and technical staff are male. As a result, the distributions of salaries for male and female employees may be quite different. The table below gives the frequencies and relative frequencies for women and men. 

(a) Explain why the total for women is greater than $100\%$

(b) Make histograms for these data, choosing the vertical scale that is most appropriate for comparing the two distributions.

(c) Write a few sentences comparing the salary distributions for men and women.

Population pyramids A population pyramid is a helpful graph for examining the distribution of a country’s population. Here is a population pyramid for Vietnam in the year $2010$. Describe what the graph tells you about Vietnam’s population that year. Be specific.

Comparing AP scores The table below gives the distribution of grades earned by students taking the AP Calculus AB and AP Statistics exams in $2009$

(a) Make an appropriate graphical display to compare the grade distributions for AP Calculus AB and AP Statistics.

(b) Write a few sentences comparing the two distributions of exam grades.

Population pyramids Refer to Exercise $64$. Here is a graph of the projected population distribution for China in the year $2050$. Describe what the graph suggests about China’s future population. Be specific. 

Student survey A survey of a large high school class asked the following questions:

(i) Are you female or male? (In the data, male = $0$, female = $1$)

(ii) Are you right-handed or left-handed? (In the data, right = $0$, left = $1$)

(iii) What is your height in inches?

(iv) How many minutes do you study on a typical weeknight?

The figure below shows histograms of the student responses, in scrambled order and without scale markings. Which histogram goes with each variable? Explain your reasoning.

Choose a graph What type of graph or graphs would you plan to make in a study of each of the following issues at your school? Explain your choices.

(a) Which radio stations are most popular with students?

(b) How many hours per week do students study?

(c) How many calories do students consume per day?

Here are the amounts of money (cents) in coins carried by 10 students in a statistics class:  $50, 35, 0, 97, 76, 0, 0, 87, 23, 65.$To make a stemplot of these data, you would use stems

(a) $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$

(b) $0, 2, 3, 5, 6, 7, 8, 9$

(c) $0, 3, 5, 6, 7.$

(d) $00, 10, 20, 30, 40, 50, 60, 70, 80, 90.$

(e) None of these.

One of the following $12$ scores was omitted from the stem plot below:$84 76 92 92 88 96 68 80 92 88 76 96$

The missing number is $\left(a\right) 76. \left(b\right) 88. \left(c\right) 90. \left(d\right) 92. \left(e\right) 96.$

One  of the following

You look at real estate ads for houses in Naples, Florida. There are many houses ranging from $\backslash (200,000 \mathrm{to} \backslash )500,000$ in price. The few houses on the water, however, have prices up to $\$15 million$. The distribution of house prices will be

(a) skewed to the left. 

(b) roughly symmetric. 

(c) skewed to the right. 

(d) unimodal. 

(e) too high. 

You look at real estate ads for 

The histogram below shows the distribution of the percent of women aged$15$ and over who have never married in each of the $50$ states and the District of Columbia. 

The leftmost bar in the histogram covers percent of never-married women ranging from about 

(a)$20\% to 24\%.$

(d) $0\% to 5\%$

(b) $20\% to 22\%$

(e) None of these. 

(c) $0\%$ 

The histogram below shows the distribution of the per cent of

women aged 15 and over who have never married in each

of the 50 states and the District of Columbia. 

The leftmost bar in the histogram covers per cent of never-married women ranging from about 

(a) $20\%$ to $24\%$

(b)$20\%$ to $22\%$

(c) $0\%$ 

(d) $0\%$ to $5\%$ 

(e) None of these.