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The Practice of Statistics for AP*

Daren S. Starnes, Daniel S. Yates, David S. Moore

Chapter 3

Describing Relationships

Educators

+ 4 more educators

Problem 1

Coral reefs How sensitive to changes in water temperature are coral reefs? To find out, measure
the growth of corals in aquariums where the water temperature is controlled at different levels.
Growth is measured by weighing the coral before and after the experiment. What are the explanatory and
response variables? Are they categorical or quantitative?

Kami D.
Numerade Educator

Problem 2

Treating breast cancer Early on, the most common treatment for breast cancer was removal of the breast. It is now usual to remove only the tumor and nearby lymph nodes, followed by radiation. The change in policy was due to a large medical experiment that compared the two treatments. Some breast cancer patients, chosen at random, were given one or the other treatment. The patients were closely followed to see how long they lived following surgery. What are the explanatory and response variables? Are they categorical or quantitative?

Ihuoma T.
Numerade Educator

Problem 3

IQ and grades Do students with higher IQ test scores tend to do better in school? The figure below shows a scatterplot of 1$Q$ and school grade point average $(\mathrm{GPA})$ for all 78 seventh-grade students in a rural midwestern school. (GPA was recorded on a 12 -point scale with $\mathrm{A}+=12, \mathrm{A}=11, \mathrm{A}-=10, \mathrm{B}+=9, \ldots,$ $\mathrm{D}-=1,$ and $\mathrm{F}=0 . )^{2}$

Kami D.
Numerade Educator

Problem 4

How much gas? Joan is concerned about the amount of energy she uses to heat her home. The graph
below plots the mean number of cubic feet of gas per day that Joan used each month against the average temperature that month (in degrees Fahrenheit) for one heating season.

Kristin C.
Numerade Educator

Problem 5

Outsourcing by airlines Airlines have increasingly outsourced the maintenance of their planes to
other companies. Critics say that the maintenance may be less carefully done, so that outsourcing creates a safety hazard. As evidence, they point to government data on percent of major maintenance out sourced and percent of flight delays blamed on the airline (often due to maintenance problems): Make a scatterplot by hand that shows how delays relate to outsourcing.3

Kami D.
Numerade Educator

Problem 6

Bird colonies One of nature's patterns connects the percent of adult birds in a colony that return from the
previous year and the number of new adults that join the colony. Here are data for 13 colonies of sparrow- hawks: Make a scatterplot by hand that shows how the number of new adults relates to the percent of returning birds.

Kami D.
Numerade Educator

Problem 7

Outsourcing by airlines Refer to your graph from Exercise $5 .$
(a) Describe the direction, form, and strength of the relationship between maintenance outsourcing and
delays blamed on the airline.
(b) One airline is a high outlier in delay percent. Which airline is this? Aside from the outlier, does the
plot show a roughly linear form? Is the relationship very strong?

Kami D.
Numerade Educator

Problem 8

Bird colonies Refer to your graph from Exercise 6.
(a) Describe the direction, form, and strength of the relationship between number of new sparrowhawks in a colony and percent of retuming adults.
(b) For short-lived birds, the association between these variables is positive; changes in weather and food supply drive the populations of new and returning birds up or down together. For long-lived territorial birds, on the other hand, the association is negative because returning birds claim their territories in the colony and don't leave room for new recruits. Which type of species is the sparrowhawk? Explain.

Kami D.
Numerade Educator

Problem 9

Does fast driving waste fuel? How does the fuel consumption of a car change as its speed increases? Here are data for a British Ford Escort. Speed is measured in kilometers per hour, and fuel consumption is measured in liters of gasoline used per 100 kilometers traveled.
(a) Make a scatterplot on your calculator.
(b) Describe the form of the relationship. Why is it not linear? Explain why the form of the relationship
makes sense.
(c) It does not make sense to describe the variables as either positively associated or negatively associated. Why?
(d) Is the relationship reasonably strong or quite weak? Explain your answer.

R M.
Numerade Educator

Problem 10

Do heavier people burn more energy? Metabolic rate, the rate at which the body consumes energy, is
important in studies of weight gain, dieting, and exercise. We have data on the lean body mass and resting metabolic rate for 12 women who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate is measured in calories burned pody mass is an important searchers believe that lean body mass is an important influence on metabolic rate. (a) Make a scatterplot on your calculator to examine the researchers' belief.
(b) Describe the direction, form, and strength of the relationship.

R M.
Numerade Educator

Problem 11

Southern education For a long time, the South has lagged behind the rest of the United States in the perr
formance of its schools. Efforts to improve education have reduced the gap. We wonder if the South stands. out in our study of state average SAT Math scores.
(a) What does the graph suggest about the southern states?
(b) The point for West Virginia is labeled in the graph. Explain how this state is an outlier.

R M.
Numerade Educator

Problem 12

Do heavier people burn more energy? The study of dieting described in Exercise 10 collected data
on the lean body mass (in kilograms) and metabolic rate (in calories) for 12 female and 7 male subjects.
The figure below is a scatterplot of the data for all 19 subjects, with separate symbols for males and females. Does the same overall pattem hold for both women and men? What is the most important difference between the sexes?

R M.
Numerade Educator

Problem 13

Merlins breeding The percent of an animal species in the wild that survives to breed again is often lower
following a successful breeding season. A study of merlins (small falcons) in northem Sweden observed
the number of breeding pairs in an isolated area and the number of breeding pairs in an isolated area and
the percent of males (banded for identification) that returned the next breeding season. Here are data for
nine years: $^{6}$ the number of breeding pairs in an isolated area and the percent of males (banded for identification) that returned the next breeding season. Here are data for nine years: $^{6}$

R M.
Numerade Educator

Problem 14

Does social rejection hurt? We often describe our emotional reaction to social rejection as "pain." Does
social rejection cause activity in areas of the brain that are known to be activated by physical pain? If
it does, we really do experience social and physical pain in similar ways. Psychologists first included and then deliberately excluded individuals from a social activity while they measured changes in brain activity. After each activity, the subjects filled out questionnaires that assessed how excluded they felt. The table below shows data for 13 subjects. "Social distress" is measured by each subject's questionnaire score after exclusion relative to the score after inclusion. (So values greater than 1 show the degree of distress caused by exclusion.) "Brain activity" is the change in activity in a region of the brain that is activated by physical pain. (So positive values show more pain.)

R M.
Numerade Educator

Problem 15

Matching correlations Five scatterplots are shown below. Match each graph to the $r$ below that best
describes it. (Some $r^{\prime}$ s will be left over.)
$$
\begin{array}{l}{r=-0.9 \quad r=-0.7 \quad r=-0.3 \quad r=0} \\ {r=0.3 \quad r=0.7 \quad r=0.9}\end{array}
$$

R M.
Numerade Educator

Problem 16

Rank the correlations Consider each of the following relationships: the heights of fathers and the heights
of their adult sons, the heights of husbands and the heights of their wives, and the heights of women at
age 4 and their heights at age $18 .$ Rank the correlations between these pairs of variables from highest to lowest. Explain your reasoning.

Andre M.
Numerade Educator

Problem 17

Correlation blunders Fach of the following statements contains an error. Explain what's wrong in each case.
(a) There is a high correlation between the gender of American workers and their income."
(b) "We found a high correlation $(r=1.09)$ between students' ratings of faculty teaching and ratings made by other faculty members."
(c) The correlation between planting rate and yield of corn was found to be $r=0.23$ bushel."

R M.
Numerade Educator

Problem 18

Teaching and research A college newspaper interviews a psychologist about student ratings of
the teaching of faculty members. The psychologist says, “The evidence indicates that the correlation be-tween the research productivity and teaching rating of faculty members is close to zero.” The paper reports this as “Professor McDaniel said that good researchers tend to be poor teachers, and vice versa.” Explain why the paper’s report is wrong. Write a statement in plain language (don’t use the word “correlation”) to explain the psychologist’s meaning.

R M.
Numerade Educator

Problem 19

Dem bones Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known. Because these specimens differ greatly in size, some scientists think they are different species rather than individuals from the same species. We will examine some data. If the specimens belong to the same species and differ in size because some are younger than others, there should be a positive linear relationship between the lengths of a pair of bones from all individuals. An outlier from this relationship would suggest a different species. Here are data on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for
the five specimens that preserve both bones:8
(a) Make a scatterplot. Do you think that all five specimens come from the same species? Explain.
(b) Find the correlation $r$ step-by-step. First, find the mean and standard deviation of each variable. Then find the six standardized values for each variable. Finally, use the formula for $r$ . Explain how your value for $r$ matches vour graph in (a).

R M.
Numerade Educator

Problem 20

Data on dating A student wonders if tall women tend to date taller men than do short women. She measures herself, her dormitory roommate, and the women in the adjoining rooms. Then she measures the next man each woman dates. Here are the data (heights in inches):
(a) Make a scatterplot of these data. Based on the scatterplot, do you expect the correlation to be positive or negative? Near $\pm 1$ or not?
(b) Find the correlation $r$ step-by-step. First, find the mean and standard deviation of each variable. Then find the six standardized values for each variable. Finally, use the formula for $r .$ Do the data show that taller women tend to date taller men?

R M.
Numerade Educator

Problem 21

Hot dogs Are hot dogs that are high in calories also high in salt? The figure below is a scatterplot of the
calories and salt content (measured as milligrams of sodium) in 17 brands of meat hot dogs.
(a) The correlation for these data is r = 0.87. Explainwhat this value means.
(b) What effect would removing the hot dog brand with the lowest calorie content have on the correlation? Justify your answer.

R M.
Numerade Educator

Problem 22

All brawn? The figure below plots the average brain weight in grams versus average body weight in kilo-
grams for 96 species of mammals. $^{10}$ There are many small mammals whose points at the lower left overlap.

R M.
Numerade Educator

Problem 23

Dem bones Refer to Exercise 19
(a) How would $r$ change if the bones had been measured in millimeters instead of centimeters? (There
are 10 millimeters in a centimeter.
(b) If the $x$ and $y$ variables are reversed, how would the correlation change? Explain.

John W.
Numerade Educator

Problem 24

Data on dating Refer to Exercise $20 .$
(a) How would $r$ change if all the men were 6 inches shorter than the heights given in the table? Does the correlation tell us if women tend to date men taller than themselves?
(b) If heights were measured in centimeters rather than inches, how would the correlation change?
(There are 2.54 centimeters in an inch.)

R M.
Numerade Educator

Problem 25

What affects correlation? Make a scatterplot of the following data: The correlation for these data is $0.5 .$ What is respon- sible for reducing the correlation to this value despite a strong straight-line relationship between $x$ and $y$ in most of the observations?

R M.
Numerade Educator

Problem 26

Strong association but no correlation The gas mileage of an automobile first increases and then decreases as the speed increases. Suppose that this relationship is very regular, as shown by the following data on speed (miles per hour) and mileage (miles per gallon). Make a scatterplot of mileage versus speed.
$$
\begin{array}{llll}{\text { Speed: }} & {20} & {30} & {40} & {50} & {60} \\ {\text { Mileage: }} & {24} & {28} & {30} & {28} & {24}\end{array}
$$
The correlation between speed and mileage is $r=0$ . Explain why the correlation is 0 even though there is a strong relationship between speed and mileage.

R M.
Numerade Educator

Problem 27

You have data for many years on the average price of a barrel of oil and the average retail price of a gallon of unleaded regular gasoline. If you want to see how well the price of oil predicts the price of gas, then you should make a scatterplot with story variable.
(a) the price of oil
(b) the price of gas
(c) the year
(d) either oil price or gas price
(e) time

R M.
Numerade Educator

Problem 28

In a scatterplot of the average price of a barrel of oil and the average retail price of a gallon of gas, you
expect to see
(a) very little association.
(b) a weak negative association.
(c) a strong negative association.
(d) a weak positive association.
(e) a strong positive association.

R M.
Numerade Educator

Problem 29

The graph at top right plots the gas mileage (miles pergallon) of various cars from the same model year versus the weight of these cars in thousands of pounds. The points marked with red dots correspond to cars made in Japan. From this plot, we may conclude that
(a) there is a positive association between weight and gas mileage for Japanese cars.
(b) the correlation between weight and gas mileage for all the cars is close to $1 .$
(c) there is little difference between Japanese cars and cars made in other countries. (d) Japanese cars tend to be lighter in weight than other cars.
(e) Japanese cars tend to get worse gas mileage than other cars.

R M.
Numerade Educator

Problem 30

If women always married men who were 2 years older than themselves, what would the correlation between the ages of husband and wife be?
$$\begin{array}{ll}{\text { (a) } 2} & {\text { (c) } 0.5} & {\text { (e) Can't tell without }} \\ {\text { (b) } 1} & {\text { (d) } 0} & {\text { seeing the data }}\end{array}$$

Richard M.
Numerade Educator

Problem 31

The figure below is a scatterplot of reading test scores against 1$Q$ test scores for 14 fifth-grade children. There is one low outlier in the plot. The 1$Q$ and reading scores for this child are
(a) $1 Q=10,$ reading $=124$
(b) $1 Q=96,$ reading $=49$
(c) $1 Q=124,$ reading $=10$ .
(d) $1 Q=145,$ reading $=100$ .
(e) $1 Q=125,$ reading $=54$ .

R M.
Numerade Educator

Problem 32

If we leave out the low outlier, the correlation for the remaining 13 points in the figure above is closest to
$$
\begin{array}{ll}{\text { (a) }-0.95} & {\text { (c) 0. (e) } 0.95}\end{array}
$$
$$
\quad \text { (b) }-0.5 . \quad \text { (d) } 0.5
$$

R M.
Numerade Educator

Problem 33

Big diamonds $(1.2,1.3)$ Here are the weights (in milligrams) of 58 diamonds from a nodule carried up
to the earth's surface in surrounding rock. These data represent a single population of diamonds formed in a single event deep in the earth."
$$
\begin{array}{cccccccccc}{13.8} & {3.7} & {33.8} & {11.8} & {27.0} & {18.9} & {19.3} & {20.8} & {25.4} & {23.1} & {7.8} \\ {10.9} & {9.0} & {9.0} & {14.4} & {6.5} & {7.3} & {5.6} & {18.5} & {1.1} & {11.2} & {7.0} \\ {7.6} & {9.0} & {9.5} & {7.7} & {7.6} & {3.2} & {6.5} & {5.4} & {7.2} & {7.2} & {3.5} \\ {5.4} & {5.1} & {5.3} & {3.8} & {2.1} & {2.1} & {4.7} & {3.7} & {3.8} & {4.9} & {2.4} \\ {1.4} & {0.1} & {4.7} & {1.5} & {2.0} & {0.1} & {0.1} & {1.6} & {3.5} & {3.7} & {2.6} \\ {4.0} & {2.3} & {4.5}\end{array}
$$
Make a graph that shows the distribution of weights of these diamonds. Describe the shape of the distribution and any outliers. Use numerical measures appropriate for the shape to describe the center and spread.

Norman A.
Numerade Educator

Problem 34

Student loans $(2.2)$ A government report looked at the amount borrowed for college by students
who graduated in 2000 and had taken out student loans. 12 The mean amount was $\overline{x}=\$ 17,776$ and the standard deviation was $s_{x}=\$ 12,034$ . The median was $\$ 15,532$ and the quartiles were $Q_{1}=\$ 9900$ and $Q_{3}=\$ 22,500$
(a) Compare the mean and the median. Also compare the distances of $Q_{1}$ and $Q_{3}$ from the median. Explain why both comparisons suggest that the distribution is right-skewed.
(b) The right-skew pulls the standard deviation up. So a Normal distribution with the same mean and
standard deviation would have a third quartile larger than the actual $Q_{3}$ . Find the third quartile of the Normal distribution with $\mu=\$ 17,776$ and $\sigma=\$ 12,034$ and compare it with $Q_{3}=\$ 22,500$ .

R M.
Numerade Educator