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Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

Chapter 9

Re-expressing Data: Get It Straight - all with Video Answers

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Chapter Questions

01:38

Problem 1

Every year, US News and World Report publishes a special issue on many U.S. colleges and universities. The scatterplots below have Student/Faculty Ratio (number of students per faculty member) for the colleges and universities on the $y$ -axes plotted against 4 other variables. The correct correlations for these scatterplots appear in this list. Match them.
$\begin{array}{llllll}-0.98 & -0.71 & -0.51 & 0.09 & 0.23 & 0.69\end{array}$

James Kiss
James Kiss
Numerade Educator
01:32

Problem 2

Are good grades in high school associated with family togetherness? A random sample of 142 high school students was asked how many meals per week their families ate together. Their responses produced a mean of 3.78 meals per week, with a standard deviation of $2.2 .$ Researchers then matched these responses against the students' grade point averages (GPAs). The scatterplot appeared to be reasonably linear, so they created a line of regression. No apparent pattern emerged in the residuals plot. The equation of the line was $\widehat{G P A}=2.73+0.11$ Meals.
a) Interpret the $y$ -intercept in this context.
b) Interpret the slope in this context.
c) What was the mean GPA for these students?
d) If a student in this study had a negative residual, what did that mean?
e) Upon hearing of this study, a counselor recommended that parents who want to improve the grades their children get should get the family to eat together more often. Do you agree with this interpretation? Explain.

James Kiss
James Kiss
Numerade Educator
02:26

Problem 3

Here are the scatterplot and regression analysis for Case Prices of 36 wines from vineyards in the Finger Lakes region of New York State and the Ages of the vineyards. (Data in Vineyards full): a) Does it appear that vineyards in business longer get higher prices for their wines? Explain.
b) What does this analysis tell us about vineyards in the rest of the world?
c) Write the regression equation.
d) Explain why that equation is essentially useless.

James Kiss
James Kiss
Numerade Educator
02:15

Problem 4

Instead of Age, perhaps the Size of the vineyard (in acres) is associated with the price of the wines. Look at the scatterplot (Data in Vineyards full): a) Do you see any evidence of an association?
b) What concern do you have about this scatterplot?
c) If the red "+" data point is removed, would the correlation become stronger or weaker? Explain.
d) If the red " $+$ " data point is removed, would the slope of the line increase or decrease? Explain.

James Kiss
James Kiss
Numerade Educator
02:42

Problem 5

In January $2012,$ the New York Times published a story called Twin Births in the U.S., Like Never Before, in which they reported a 76 percent increase in the rate of twin births from 1980 to $2009 .$ Here are the number of twins born each year (per 1000 live births). (www.cdc.gov/nchs/births.htm)
a) Using the data only up to 2009 (as in the NYTimes article), find the equation of the regression line for predicting the number of twin births by Years since 1980 .
b) Explain in this context what the slope means.
c) Predict the number of twin births in the United States for the year 2014 . Comment on your faith in that prediction.
d) Comment on the residuals plot and the appropriateness of the model. Look at the past three years' rates.

Nick Johnson
Nick Johnson
Numerade Educator
03:48

Problem 6

The Dow Jones stock index measures the performance of the stocks of America's largest companies (finance.yahoo.com). A regression of the Dow prices on years $1972-2014$ looks like this:
Dependent variable is Dow Index R-squared $=88.3 \% \quad s=1604$
Variable Coefficient Intercept -2485.66
Year since 1 1970
354.24
a) What is the correlation between $D$ ow Index and Year?
b) Write the regression equation.
c) Explain in this context what the equation says.
d) Here's a scatterplot of the residuals. Comment on the appropriateness of the model.

Teresa Wray
Teresa Wray
Numerade Educator
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Problem 7

Biologists studying the effects of acid rain on wildlife collected data from 163 streams in the Adirondack Mountains. They recorded the $p H$ (acidity) of the water and the $B C I$, a measure of biological diversity, and they calculated $R^{2}=27 \% .$ Here's a scatterplot of $B C I$ against $p H:$
a) What is the correlation between $p H$ and $B C I ?$
b) Describe the association between these two variables.
c) If a stream has average $p H,$ what would you predict about the $B C I ?$
d) In a stream where the $p H$ is 3 standard deviations above average, what would you predict about the $B C I ?$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:31

Problem 8

Marine biologists warn that the growing number of powerboats registered in Florida threatens the existence of manatees. The data below come from the Florida Fish and Wildlife Conservation Commission (myfwc.com/research/manatee/) and the National Marine Manufacturers Association (www.nmma.org/).
a) In this context, which is the explanatory variable?
b) Make a scatterplot of these data and describe the association you see.
c) Find the correlation between Boat Registrations and Manatee Deaths.
d) Interpret the value of $R^{2}$.
e) Does your analysis prove that powerboats are killing manatees?

Lynn Larson
Lynn Larson
Numerade Educator
02:04

Problem 9

Continue your analysis of the manatee situation from the previous exercise.
a) Create a linear model of the association between Manatee Deaths and Powerboat Registrations.
b) Interpret the slope of your model.
c) Interpret the $y$ -intercept of your model.
d) How accurately did your model predict the high number of manatee deaths in $2009 ?$
e) Which is better for the manatees, positive residuals or negative residuals? Explain.
f) What does your model suggest about the future for the manatee?

Lynn Larson
Lynn Larson
Numerade Educator
05:16

Problem 10

A Statistics instructor created a linear regression equation to predict students' final exam scores from their midterm exam scores. The regression equation was $\widehat{F i n}=10+0.9$ Mid.
a) If Susan scored a 70 on the midterm, what did the instructor predict for her score on the final?
b) Susan got an 80 on the final. How big is her residual?
c) If the standard deviation of the final was 12 points and the standard deviation of the midterm was 10 points, what is the correlation between the two tests?
d) How many points would someone need to score on the midterm to have a predicted final score of $100 ?$
e) Suppose someone scored 100 on the final. Explain why you can't estimate this student's midterm score from the information given.
f) One of the students in the class scored 100 on the midterm but got overconfident, slacked off, and scored only 15 on the final exam. What is the residual for this student?
g) No other student in the class "achieved" such a dramatic turnaround. If the instructor decides not to include this student's scores when constructing a new regression model, will the $R^{2}$ value of the regression increase, decrease, or remain the same? Explain.
h) Will the slope of the new line increase or decrease?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:23

Problem 11

Highway planners investigated the relationship between traffic Density (number of automobiles per mile) and the average Speed of the traffic on a moderately large city thoroughfare. The data were collected at the same location at 10 different times over a span of 3 months. They found a mean traffic Density of 68.6 cars per mile (cpm) with standard deviation of 27.07 cpm. Overall, the cars' average Speed was $26.38 \mathrm{mph}$, with standard deviation of 9.68 mph. These researchers found the regression line for these data to be $\widehat{\text { Speed }}=50.55-0.352$ Density.
a) What is the value of the correlation coefficient between Speed and Density?
b) What percent of the variation in average Speed is explained by traffic Density?
c) Predict the average Speed of traffic on the thoroughfare when the traffic Density is $50 \mathrm{cpm} .$
d) What is the value of the residual for a traffic Density of $56 \mathrm{cpm}$ with an observed Speed of $32.5 \mathrm{mph} ?$
e) The data set initially included the point Density = 125 cpm, Speed $=55 \mathrm{mph}$. This point was considered an outlier and was not included in the analysis. Will the slope increase, decrease, or remain the same if we redo the analysis and include this point?
f) Will the correlation become stronger, weaker, or remain the same if we redo the analysis and include this point (125,55)$?$
$\mathrm{g}$ ) A European member of the research team measured the Speed of the cars in kilometers per hour $(1 \mathrm{~km} \approx 0.62$ miles $)$ and the traffic Density in cars per kilometer. Find the value of his calculated correlation between speed and density.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:49

Problem 12

Thursday, researchers gave students enrolled in a section of basic Spanish a set of 50 new vocabulary words to memorize. On Friday, the students took a vocabulary test. When they returned to class the following Monday, they were retested-without advance warning. Here are the test scores for the 25 students.
$$\begin{array}{|c|c|c|c|c|c|}\hline \text { Fri. } & \text { Mon. } & \text { Fri. } & \text { Mon. } & \text { Fri. } & \text {Mon. } \\\hline 42 & 36 & 48 & 37 & 39 & 41 \\
44 & 44 & 43 & 41 & 46 & 32 \\45 & 46 & 45 & 32 & 37 & 36 \\
48 & 38 & 47 & 44 & 40 & 31 \\44 & 40 & 50 & 47 & 41 & 32 \\
43 & 38 & 34 & 34 & 48 & 39 \\41 & 37 & 38 & 31 & 37 & 31 \\
35 & 31 & 43 & 40 & 36 & 41 \\\hline 43 & 32 & & & & \\\hline\end{array}$$
a) What is the correlation between Friday and Monday scores?
b) What does a scatterplot show about the association between the scores?
c) What does it mean for a student to have a positive residual?
d) What would you predict about a student whose Friday score was one standard deviation below average?
e) Write the equation of the regression line.
f) Predict the Monday score of a student who earned a 40 on Friday.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
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Problem 13

What factor most explains differences in Fuel Efficiency among cars? Below is a correlation matrix exploring that relationship for the car's Weight, Horsepower, engine size (Displacement), and number of Cylinders. (Data in Cars)
a) Which factor seems most strongly associated with Fuel Efficiency?
b) What does the negative correlation indicate?
c) Explain the meaning of $R^{2}$ for that relationship.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 14

Look again at the correlation table for cars in the previous exercise.
a) Which two variables in the table exhibit the strongest association?
b) Is that strong association necessarily cause-and-effect? Offer at least two explanations why that association might be so strong.
c) Engine displacements for U.S.-made cars are often measured in cubic inches. For many foreign cars, the units are either cubic centimeters or liters. How would changing from cubic inches to liters affect the calculated correlations involving Displacement?
d) What would you predict about the Fuel Efficiency of a car whose engine Displacement is one standard deviation above the mean?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:38

Problem 15

Can we predict the Horsepower of the engine that manufacturers will put in a car by knowing the Weight of the car? Here are the regression analysis and residuals plot:
a) Write the equation of the regression line.
b) Do you think the car's Weight is measured in pounds or thousands of pounds? Explain.
c) Do you think this linear model is appropriate? Explain.
d) The highest point in the residuals plot, representing a residual of 22.5 horsepower, is for a Chevy weighing 2595 pounds. How much horsepower does this car have?

Teresa Wray
Teresa Wray
Numerade Educator
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Problem 16

Although some women are colorblind, this condition is found primarily in men. Why is it wrong to say there's a strong correlation between $\operatorname{Sex}$ and Colorblindness?

Bryan Meares
Bryan Meares
Numerade Educator
03:26

Problem 17

There is evidence that eruptions of Old Faithful can best be predicted by knowing the duration of the previous eruption.
a) Describe what you see in the scatterplot of $\underline{\text { Intervals }}$ between eruptions vs. Duration of the previous eruption.
b) Write the equation of the line of best fit. Here's the regression analysis:
Dependent variable is Interval R-squared $=77.0 \% \quad \mathrm{~s}=6.16 \mathrm{~min}$
Variable Coefficient
Intercept 33.9668
Duration 10.3582
c) Carefully explain what the slope of the line means in this context.
d) How accurate do you expect predictions based on this model to be? Cite statistical evidence.
e) If you just witnessed an eruption that lasted 4 minutes, how long do you predict you'll have to wait to see the next eruption?
f) So you waited, and the next eruption came in 79 minutes. Use this as an example to define a residual.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:33

Problem 18

The ranges inhabited by the Indian gharial crocodile and the Australian saltwater crocodile overlap in Bangladesh. Suppose a very large crocodile skeleton is found there, and we wish to determine the species of the animal. Wildlife scientists have measured the lengths of the heads and the complete bodies of several crocs (in centimeters) of each species, creating the regression analyses below:
a) Do the associations between the sizes of the heads and bodies of the two species appear to be strong? Explain.
b) In what ways are the two relationships similar? Explain.
c) What is different about the two models? What does that mean?
d) The crocodile skeleton found had a head length of $62 \mathrm{~cm}$ and a body length of $380 \mathrm{~cm}$. Which species do you think it was? Explain why.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:30

Problem 19

One can determine how old a tree is by counting its rings, but that requires either cutting the tree down or extracting a sample from the tree's core. Can we estimate the tree's age simply from its diameter? A forester measured 27 trees of the same species that had been cut down, and counted the rings to determine the ages of the trees.
$$\begin{array}{|c|c|c|c|}\hline \text { Diameter (in.) } & \text { Age (yr) } & \text { Diameter (in.) } & \text { Age (yr) } \\
\hline 1.8 & 4 & 10.3 & 23 \\1.8 & 5 & 14.3 & 25 \\2.2 & 8 & 13.2 & 28 \\4.4 & 8 & 9.9 & 79 \\
6.6 & 8 & 13.2 & 30 \\4.4 & 10 & 15.4 & 30 \\7.7 & 10 & 17.6 & 33 \\10.8 & 12 & 14.3 & 34 \\7.7 & 13 & 15.4 & 35 \\5.5 & 14 & 11.0 & 38 \\9.9 & 16 & 15.4 & 38 \\10.1 & 18 & 16.5 & 40 \\
12.1 & 20 & 16.5 & 42 \\12.8 & 22 & & \\\hline\end{array}$$
a) Find the correlation between Diameter and Age. Does this suggest that a linear model may be appropriate? Explain.
b) Create a scatterplot and describe the association.
c) Create the linear model.
d) Check the residuals. Explain why a linear model is probably not appropriate.
e) If you used this model, would it generally overestimate or underestimate the ages of very large trees? Explain.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:04

Problem 20

In the last exercise, you saw that the linear model had some deficiencies. Let's create a better model.
a) Perhaps the cross-sectional area of a tree would be a better predictor of its age. Since area is measured in square units, try re-expressing the data by squaring the diameters. Does the scatterplot look better?
b) Create a model that predicts Age from the square of the Diameter.
c) Check the residuals plot for this new model. Is this model more appropriate? Why?
d) Estimate the age of a tree 18 inches in diameter.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:40

Problem 21

An electronics website collects data on the size of new HD flat panel televisions (measuring the diagonal of the screen in inches) to predict the cost (in hundreds of dollars). Which of these is most likely to be the slope of the regression line: $0.03,0.3,3,30 ?$ Explain.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:30

Problem 22

The Child Trends Data Bank monitors issues related to children. The table shows
a 50 -state average of the percent of expectant mothers who smoked cigarettes during their pregnancies.
$$\begin{array}{|c|c|c|c|}\hline & \text { \% Smoking } & & \text { \% Smoking } \\
\text { Year } & \text { While Pregnant } & \text { Year } & \text { While Pregnant } \\\hline 1990 & 19.2 & 2001 & 13.8 \\1991 & 18.7 & 2002 & 13.3 \\
1992 & 17.9 & 2003 & 12.7 \\
1993 & 16.8 & 2004 & 10.9 \\
1994 & 16.0 & 2005 & 10.1 \\
1995 & 15.4 & 2006 & 10.0 \\
1996 & 15.3 & 2007 & 10.4 \\
1997 & 14.9 & 2008 & 9.7 \\
1998 & 14.8 & 2009 & 9.3 \\
1999 & 14.1 & 2010 & 9.2 \\
2000 & 14.0 & 2011 & 9.0 \\\hline\end{array}$$
a) Create a scatterplot and describe the trend you see.
b) Find the correlation.
c) How is the value of the correlation affected by the fact that the data are averages rather than percentages for each of the 50 states?
d) Write a linear model and interpret the slope in context.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
View

Problem 23

The downward trend in smoking you saw in the last exercise is good news for the health of babies, but will it ever stop?
a) Explain why you can't use the linear model you created in Exercise 22 to see when smoking during pregnancy will cease altogether.
b) Create a model that could estimate the year in which the level of smoking would be $0 \%$.
c) Comment on the reliability of such a prediction.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 24

It's commonly believed that people use tips to reward good service. A researcher for the hospitality industry examined tips and ratings of service quality from 2645 dining parties at 21 different restaurants. The correlation between ratings of service and tip percentages was 0.11. (Source: M. Lynn and M. McCall, "Gratitude and Gratuity." Journal of Socio-Economics $29: 203-214)$
a) Describe the relationship between $Q$ uality of Service and Tip Size.
b) Find and interpret the value of $R^{2}$ in this context.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 25

Data from 50 large U.S. cities show the mean January Temperature and the Latitude. Describe what you see in the scatterplot.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
04:30

Problem 26

The study of U.S. cities in Exercise 25 found the mean January Temperature (degrees Fahrenheit), Altitude (feet above sea level), and Latitude (degrees north of the equator) for 55 cities. Here's the correlation matrix:
$$\begin{aligned}
&\begin{array}{l|l|r}\text { Variable } & \text { Mean } & \text { StdDev } \\
\text { Latitude } & 39.02 & 5.42 \\
\text { JanTemp } & 26.44 & 13.49
\end{array}\\&\text { Correlation }=-0.848\end{aligned}$$
a) What percent of the variation in January Temperature can be explained by variation in Latitude?
b) What is indicated by the fact that the correlation is negative?
c) Write the equation of the line of regression for predicting average January Temperature from Latitude.
d) Explain what the slope of the line means.
e) Do you think the $y$ -intercept is meaningful? Explain.
f) The latitude of Denver is $40^{\circ} \mathrm{N}$. Predict the mean January temperature there.
g) What does it mean if the residual for a city is positive?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:39

Problem 27

The value of a log is based on the number of board feet of lumber the log may contain. (A board foot is the equivalent of a piece of wood I inch thick, 12 inches wide, and 1 foot long. For example, a $2^{\prime \prime} \times 4^{\prime \prime}$ piece that is 12 feet long contains 8 board feet.) To estimate the amount of lumber in a log, buyers measure the diameter inside the bark at the smaller end. Then they look in a table based on the Doyle Log Scale. The table below shows the estimates for logs 16 feet long.
$$
\begin{array}{|l|cccccc|}
\hline \text { Diameter of Log } & 8^{\prime \prime} & 12^{*} & 16^{*} & 20^{*} & 24^{*} & 28^{*} \\
\text { Board Feet } & 16 & 64 & 144 & 256 & 400 & 576 \\
\hline
\end{array}
$$
a) What model does this scale use?
b) How much lumber would you estimate that a $\log 10$ inches in diameter contains?
c) What does this model suggest about logs 36 inches in diameter?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:49

Problem 28

The September 1998 issue of the American Psychologist published an article by Kraut et al. that reported on an experiment examining "the social and psychological impact of the Internet on 169 people in 73 households during their first 1 to 2 years online." In the experiment, 73 households were offered free Internet access for 1 or 2 years in return for allowing their time and activity online to be tracked. The members of the households who participated in the study were also given a battery of tests at the beginning and again at the end of the study. The conclusion of the study made news headlines: Those who spent more time online tended to be more depressed at the end of the experiment. Although the paper reports a more complex model, the basic result can be summarized in the following regression of Depression (at the end of the study, in "depression scale units") vs. Internet $U$ se (in mean hours per week):
Dependent variable is Depression R-squared $=4.6 \% \quad s=0.4563$
$\begin{array}{ll}\text { Variable } & \text { Coefficient }\end{array}$ Intercept 0.5655
Internet use
0.0199
The news reports about this study clearly concluded that using the Internet causes depression. Discuss whether such a conclusion can be drawn from this regression. If so, discuss the supporting evidence. If not, say why not.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:31

Problem 29

How are Olympic performances in various events related? The plot shows winning long-jump and high-jump distances, in meters, for the Summer Olympics from 1912 through $2012 .$ a) Describe the association.
b) Do long-jump performances somehow influence the high-jumpers? How do you account for the relationship you see?
c) The correlation for the given scatterplot is 0.912 . If we converted the jump lengths to centimeters by multiplying by 100 , would that make the actual correlation higher or lower?
d) What would you predict about the long jump in a year when the high-jumper jumped one standard deviation better than the average high jump?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:22

Problem 30

Here are the summary statistics for the Olympic long jumps and high jumps displayed in the previous exercise.
$$\begin{array}{l|c|c}\text { Event } & \text { Mean } & \text { StdDev } \\
\text { High Jump } & 2.148077 & 0.1938663 \\
\text { Long Jump } & 8.050000 & 0.5135913 \\
{\text { Correlation }=0.91246}\end{array}$$
a) Write the equation of the line of regression for estimating High Jump from Long Jump.
b) Interpret the slope of the line.
c) In a year when the long jump is $8.9 \mathrm{~m}$, what high jump would you predict?
d) Why can't you use this line to estimate the long jump for a year when you know the high jump was $2.25 \mathrm{~m} ?$
e) Write the equation of the line you need to make that prediction.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
09:46

Problem 31

Consider the association between a student's score on a French vocabulary test and the weight of the student. What direction and strength of correlation would you expect in each of the following situations? Explain.
a) The students are all in third grade.
b) The students are in third through twelfth grades in the same school district.
c) The students are in tenth grade in France.
d) The students are in third through twelfth grades in France.

Lucía Guerrero
Lucía Guerrero
Numerade Educator
01:59

Problem 32

are often born at less than 9 months gestation. The graph from the Journal of the American Medical Association (JAMA) shows the rate of preterm twin births in the United States over the past 20 years. In this study, JAMA categorized mothers by the level of prenatal medical care they received: inadequate, adequate, or intensive.
a) Describe the overall trend in preterm twin births.
b) Describe any differences you see in this trend, depending on the level of prenatal medical care the mother received.
c) Should expectant mothers be advised to cut back on the level of medical care they seek in the hope of avoiding preterm births? Explain.

Nick Johnson
Nick Johnson
Numerade Educator
01:42

Problem 33

Does how long toddlers sit at the lunch table help predict how much they eat? The table and graph show the number of minutes the kids stayed at the table and the number of calories they consumed. Create and interpret a model for these data. $$\begin{array}{|c|c|c|c|}\hline \text { Calories } & \text { Time } & \text { Calories } & \text { Time } \\\hline 472 & 21.4 & 450 & 42.4 \\498 & 30.8 & 410 & 43.1 \\465 & 37.7 & 504 & 29.2 \\456 & 33.5 & 437 & 31.3 \\
423 & 32.8 & 489 & 28.6 \\437 & 39.5 & 436 & 32.9 \\508 & 22.8 & 480 & 30.6 \\
431 & 34.1 & 439 & 35.1 \\479 & 33.9 & 444 & 33.0 \\454 & 43.8 & 408 & 43.7 \\\hline\end{array}$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:29

Problem 34

Since clean-air regulations have dictated the use of unleaded gasoline, the supply of leaded gas in New York state has diminished. The following table was given on the August 2001 New York State Math $\mathrm{B}$ exam, a statewide achievement test for high school students.$$\begin{array}{|l|l|l|l|l|l|}\hline \text { Year } & 1984 & 19888 & 1992 & 1996 & 2000 \\\hline \text { Gallons (1000''s) } & 150 & 124 & 104 & 76 & 50 \\\hline\end{array}$$
a) Create a linear model and predict the number of gallons that will be available in $2015 .$ Comment.
b) The exam then asked students to estimate the year when leaded gasoline will first become unavailable, expecting them to use the model from part a to answer the question. Explain why that method is incorrect.
c) Create a model that would be appropriate for that task, and make the estimate.
d) The "wrong" answer from the other model is fairly accurate in this case. Why?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:39

Problem 35

Are people who use tobacco products more likely to consume alcohol? Here are data on household spending (in pounds) taken by the British government on 11 regions in Great Britain. Do tobacco and alcohol spending appear to be related? What questions do you have about these data? What conclusions can you draw?$$\begin{array}{l|c|c}
\hline \text { Region } & \text { Alcohol } & \text { Tobacco } \\
\hline \text { North } & 6.47 & 4.03 \\
\text { Yorkshire } & 6.13 & 3.76 \\
\text { Northeast } & 6.19 & 3.77 \\
\text { East Midlands } & 4.89 & 3.34 \\
\text { West Midlands } & 5.63 & 3.47 \\
\text { East Anglia } & 4.52 & 2.92 \\
\text { Southeast } & 5.89 & 3.20 \\
\text { Southwest } & 4.79 & 2.71 \\
\text { Wales } & 5.27 & 3.53 \\
\text { Scotland } & 6.08 & 4.51 \\
\text { Northern Ireland } & 4.02 & 4.56 \\
\hline\end{array}$$

Oluwadamilola Ameobi
Oluwadamilola Ameobi
Numerade Educator
02:37

Problem 36

The Sears Cup was established in 1993 to honor institutions that maintain a broad-based athletic program, achieving success in many sports, both men's and women's. Since its Division III inception in 1995 , the cup has been won by Williams College 15 of 17 years. Their football team has an $85.3 \%$ winning record under their current coach. Why does the football team win so much? Is it because they're heavier than their opponents? The table shows the average team weights for selected years from 1973 to 1993 .
$$\begin{array}{|c|c|c|c|}
\hline \text { Year } & \text { Weight (Ib) } & \text { Year } & \text { Weight (Ib) } \\
\hline 1973 & 185.5 & 1983 & 192.0 \\
1975 & 182.4 & 1987 & 196.9 \\
1977 & 182.1 & 1989 & 202.9 \\
1979 & 191.1 & 1991 & 206.0 \\
1981 & 189.4 & 1993 & 198.7 \\
\hline\end{array}$$
a) Fit a straight line to the relationship between Weight and Year.
b) Does a straight line seem reasonable?
c) Predict the average weight of the team for the year
2015. Does this seem reasonable?
d) What about the prediction for the year $2103 ?$ Explain.
e) What about the prediction for the year 3003 ? Explain.

Lucas Finney
Lucas Finney
Numerade Educator
06:47

Problem 37

Find the predicted value of $y$, using each model for $x=10$.
a) $\hat{y}=2+0.8 \ln x$
b) $\log \hat{y}=5-0.23 x$
c) $\frac{1}{\sqrt{\hat{y}}}=17.1-1.66 x$

Trinity Steen
Trinity Steen
Numerade Educator
05:05

Problem 38

Here are the average weights of the football team for the University of Texas for various years in the 20 th century.
$$\begin{array}{|l|l|l|l|l|l|l|}
\hline \text { Year } & 1905 & 1919 & 1932 & 1945 & 1955 & 1965 \\\hline \text { Weight (Ib) } & 164 & 163 & 181 & 192 & 195 & 199 \\\hline\end{array}$$
a) Fit a straight line to the relationship of Weight by Year for Texas football players.
b) According to these models, in what year will the predicted weight of the Williams College team from Exercise 36 first be more than the weight of the University of Texas team?
c) Do you believe this? Explain.

Dawn Getman
Dawn Getman
Numerade Educator
03:01

Problem 39

The Minnesota Department of Transportation hoped that they could measure the weights of big trucks without actually stopping the vehicles by using a newly developed "weigh-in-motion" scale. After installation of the scale, a study was conducted to find out whether the scale's readings correspond to the true weights of the trucks being monitored. In Chapter 6 , Exercise 50 , you examined the scatterplot for the data they collected, finding the association to be approximately linear with $R^{2}=93 \%$. Their regression equation is $\widehat{W t}=10.85+0.64$ Scale, where both the scale reading and the predicted weight of the truck are measured in thousands of pounds.
a) Estimate the weight of a truck if this scale read 31,200 pounds.
b) If that truck actually weighed 32,120 pounds, what was the residual?
c) If the scale reads 35,590 pounds, and the truck has a residual of -2440 pounds, how much does it actually weigh?
d) In general, do you expect estimates made using this equation to be reasonably accurate? Explain.
e) If the police plan to use this scale to issue tickets to trucks that appear to be overloaded, will negative or positive residuals be a greater problem? Explain.

James Kiss
James Kiss
Numerade Educator
03:27

Problem 40

How are a company's profits related to its sales? Let's examine data from 71 large U.S. corporations. All amounts are in millions of dollars.
a) Histograms of Profits and Sales and histograms of the logarithms of Profits and Sales are seen below. Why are the re-expressed data better for regression?
b) Here are the scatterplot and residuals plot for the regression of logarithm of Profits vs. log of Sales. Do you think this model is appropriate? Explain.
c) Here's the regression analysis. Write the equation.
Dependent variable is Log Profit R-squared $=48.1 \%$ Variable Coefficient
$$\begin{array}{lr}
\text { Intercept } & -0.106259 \\
\text { LogSales } & 0.647798
\end{array}$$
d) Use your equation to estimate profits earned by a company with sales of 2.5 billion dollars. (That's 2500 million.)

James Kiss
James Kiss
Numerade Educator
03:56

Problem 41

Most water tanks have a drain plug so that the tank may be emptied when it's to be moved or repaired. How long it takes a certain size of tank to drain depends on the size of the plug, as shown in the table. Create a model.
$$\begin{array}{|l|c|c|c|c|c|c|c|}
\hline \begin{array}{l}\text { Plug } \\\text { Dia (in.) }\end{array} & \frac{3}{8} & \frac{1}{2} & \frac{3}{4} & 1 & 1 \frac{1}{4} & 1 \frac{1}{2} & 2 \\\hline \begin{array}{l}\text { Drain } \\\text { Time (min.) }\end{array} & 140 & 80 & 35 & 20 & 13 & 10 & 5 \\\hline\end{array}$$

Vipender Yadav
Vipender Yadav
Numerade Educator
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Problem 42

A start-up company has developed an improved electronic chip for use in laboratory equipment. The company needs to project the manufacturing cost, so it develops a spreadsheet model that takes into account the purchase of production equipment, overhead, raw materials, depreciation, maintenance, and other business costs. The spreadsheet estimates the cost of producing 10,000 to 200,000 chips per year, as seen in the table. Develop a regression model to predict Costs based on the Level of production.
$$\begin{array}{|c|c|c|c|}\hline \begin{array}{c}\text { Chips } \\\text { Produced } \\
\text { (1000s) }\end{array} & \begin{array}{c}\text { Cost } \\\text { per } \\\text { Chip (\$) }
\end{array} & \begin{array}{c}\text { Chips } \\
\text { Produced } \\text { (1000s) }\end{array} & \begin{array}{c}\text { Cost } \\\text { per } \\\text { Chip (\$)}\end{array} \\\hline 10 & 146.10 & 90 & 47.22 \\20 & 105.80 & 100 & 44.31 \\30 & 85.75 & 120 & 42.88 \\
40 & 77.02 & 140 & 39.05 \\50 & 66.10 & 160 & 37.47 \\60 & 63.92 & 180 & 35.09 \\
70 & 58.80 & 200 & 34.04 \\80 & 50.91 & & \\\hline\end{array}$$

Carson Merrill
Carson Merrill
Numerade Educator