Deck 2: Data

Question 1

All but one of the statements below contain a mistake. Which one could be true?

A) There is a strong correlation between type of preferred pet and income level.
B) The correlation between the height of a bean plant and the day is 0.78 in/day.
C) The correlation between your golf score and the number of hours you practice is -0.36.
D) The correlation between the time it takes to get ready in the morning and gender is 0.78.
E) The number of apricots on a tree and the amount of fertilizer have a 1.12 correlation.

Answer 1

C

Question 2

A correlation of zero between two quantitative variables means that

A) re-expressing the data will guarantee a linear association between the two variables.
B) we have done something wrong in our calculation of r.
C) there is no linear association between the two variables.
D) none of these
E) there is no association between the two variables.

Answer 2

C

Question 3

It takes a while for new factory workers to master a complex assembly process. During the first month new employees work, the company tracks the number of days they have been on the job and the length of time it takes them to complete an assembly. The correlation is most likely to be

A) near 0
B) exactly -1.0
C) exactly +1.0
D) near -0.6
E) near +0.6

Answer 3

D

Question 4

All but one of the statements below contain a mistake. Which one could be true?

A) The correlation between a person's age and vision (20/20?) is r = -1.04.
B) The correlation between blood alcohol level and reaction time is r = 0.73.
C) There is a high correlation between cigarette smoking and gender.
D) The correlation between the species of tree and its height is r = 0.56.
E) The correlation between age and weight of a newborn baby is r = 0.83 ounces per day.

Question 5

Researchers studying growth patterns of children collect data on the heights of fathers and sons.
The correlation between the fathers' heights and the heights of their 16 year-old sons is most likely to be...

A) near +0.7
B) near -1.0
C) exactly +1.0
D) somewhat greater than 1.0
E) near 0

Question 6

All but one of these statements contain a mistake. Which could be true?

A) The correlation between a football player's weight and the position he plays is 0.54.
B) There is a correlation of 0.63 between gender and political party.
C) There is a high correlation (1.09) between height of a corn stalk and its age in weeks.
D) The correlation between a car's length and its fuel efficiency is 0.71 miles per gallon.
E) The correlation between the amount of fertilizer used and the yield of beans is 0.42.

Question 7

A business owner notes that for every extra hour his store is open, his total sales increase by a fixed amount. His most useful predictive model is probably……

A) exponential
B) power
C) quadratic
D) logarithmic
E) linear

Question 8

Which of the following is not a source of caution in regression analysis between two variables?

A) an outlier.
B) All of these are potential problems.
C) subgroups with different characteristics.
D) extrapolation.
E) a lurking variable.

Question 9

Two variables that are actually not related to each other may nonetheless have a very high correlation because they both result from some other, possibly hidden, factor. This is an example of

A) a lurking variable.
B) leverage.
C) an outlier.
D) regression.
E) extrapolation.

Question 10

A residual plot that has no pattern is a sign that……

A) the original data is straight and the regression line is a good model.
B) the original data is straight and the regression line is not a good model.
C) the model is not a good one, because there is no pattern.
D) the original data is curved and the regression line is a good model.
E) the original data is curved and the regression line is not a good model.

Question 11

The auto insurance industry crashed some test vehicles into a cement barrier at speeds of 5 to 25 mph to investigate the amount of damage to the cars. They found a correlation of r = 0.60 between speed (MPH) and damage ($). If the speed at which a car hit the barrier is 1.5 standard deviations above the mean speed, we expect the damage to be_______ the mean damage.

A) 0.36 SD above
B) equal to
C) 0.90 SD above
D) 1.5 SD above
E) 0.60 SD above

Question 12

An 8th grade class develops a linear model that predicts the number of cheerios (a small round cereal) that fit on the circumference of a plate by using the diameter in inches. Their model is cheerios = 0.56 + 5.11(diameter).


-The slope of this model is best interpreted in context as……

A) For every 5.11 inches of diameter, the circumference is about 1 cheerio bigger.
B) It takes 5.11 cheerios to fill a plate's circumference.
C) For every 1 inch of diameter, the circumference holds about 0.56 more cheerios.
D) A mistake, because n is about 3.14 and that should be the slope.
E) For every 1 inch of diameter, the circumference holds about 5.11 more cheerios.

Question 13

Another farmer has increased his wheat production by about the same percentage each year. His most useful predictive model is probably…

A) exponential
B) linear
C) quadratic
D) power
E) logarithmic

Question 14

If a data point is influential it……

A) is guaranteed to be extreme in the horizontal direction.
B) none of these
C) is guaranteed to be extreme in the vertical direction.
D) has a small residual.
E) will change the slope of the regression equation.

Question 15

If the point in the upper right corner of this scatterplot is removed from the data set, then what will happen to the slope of the line of best fit (b) and to the correlation (r)?

A) both will decrease.
B) both will remain the same.
C) both will increase.
D) b will decrease, and r will increase.
E) b will increase, and r will decrease.

Question 16

A consumer group collected information on HDTVs. They created a linear model to estimate the cost of an HDTV (in $) based on the screen size (in inches). Which is the most likely value of the slope of the line of best fit?

A) 7
B) 0.70
C) 700
D) 7000
E) 70

Question 17

The residuals plot for a linear model is shown. Which is true?

A) The linear model is no good because of the curve in the residuals.
B) The linear model is okay because approximately the same number of points are above the line as below it.
C) The linear model is no good because some residuals are large.
D) The linear model is no good because the correlation is near 0.
E) The linear model is okay because the association between the two variables is fairly strong.

Question 18

R-sq is a measure of ……

A) the percentage of the accuracy of the regression equation.
B) the proportion of the variability in the response variable that is explained by the explanatory variable.
C) the initial predicted starting point of the response variable when x is zero.
D) the probability that the regression line makes a correct prediction.
E) the change in the y-variable that corresponds with the change in the x-variable.

Question 19

The correlation between X and Y is r = 0.35. If we double each X value, decrease each Y by 0.20, and interchange the variables (put X on the Y-axis and vice versa), the new correlation

A) is 0.50
B) is 0.90
C) is 0.35
D) is 0.70
E) cannot be determined.

Question 20

When using midterm exam scores to predict a student's final grade in a class, the student would prefer to have a

A) residual equal to zero, because that means the student's final grade is exactly what we would predict with the model.
B) negative residual, because that means the student's final grade is lower than we would predict with the model.
C) negative residual, because that means the student's final grade is higher than we would predict with the model.
D) positive residual, because that means the student's final grade is higher than we would predict with the model.
E) positive residual, because that means the student's final grade is lower than we would predict with the model.

Question 21

Suppose we collect data hoping to be able to estimate the prices of commonly owned new cars (in
$) from their lengths (in feet). Of these possibilities, the slope of the line of best fit is most likely to
Be

A) 3000
B) 30,000
C) 3
D) 300
E) 30

Question 22

A lakeside restaurant found the correlation between the daily temperature and the number of meals they served to be 0.40. On a day when the temperature is two standard deviations above the mean, the number of meals they should plan on serving is __________the mean.

A) 0.4 SD above
B) 2.0 SD above
C) 0.16 SD above
D) 0.8 SD above
E) equal to

Question 23

The relationship between the longevity of an animal's life and its gestation time is 0.70. If an animal is one standard deviation below average in life expectancy, the gestation time is predicted to _____below average.

A) 1 SD
B) none of these
C) 1.4 SD
D) 0.7 SD
E) 0.49 SD

Question 24

A regression model examining the amount of distance a long distance runner runs (in miles) to predict the amount of fluid the runner drinks (ounces) has a slope of 4.6. Which interpretation is appropriate?

A) Each mile adds 4.6 more ounces.
B) We predict 4.6 miles for every ounce that is drunk.
C) We predict for every mile run, the runner drinks 4.6 more ounces.
D) The correlation is needed to interpret this value.
E) A runner drinks a minimum of 4.6 oz.

Question 25

Which scatterplot shows a strong association between two variables even though the correlation is probably near zero?

A)

B)

C)

D)

E)

Question 26

Over the past decade a farmer has been able to increase his wheat production by about the same number of bushels each year. His most useful predictive model is probably…

A) exponential
B) linear
C) power
D) quadratic
E) logarithmic

Question 27

All but one of the statements below contain a mistake. Which one could be true?

A) The correlation between gender and age is -0.171.
B) The correlation between weight and length of foot is 0.488.
C) The correlation between the breed of a dog and its weight is 0.435.
D) The correlation between height and weight is 0.568 inches per pound.
E) If the correlation between blood alcohol level and reaction time is 0.73, then the correlation between reaction time and blood alcohol level is -0.73.

Question 28

A regression equation is found that predicts the increased cost of a home owner's electricity bill Given the number of holiday lights they put on the outside of their house. The equation is dollars =2.5 + 0.02(light) . If a house has 400 lights and a $15 increase in their electricity cost, find their residual.

A) $20
B) $5
C) -$15
D) $15
E) -$5

Question 29

An 8th grade class develops a linear model that predicts the number of cheerios (a small round cereal) that fit on the

circumference of a plate by using the diameter in inches. Their model is $\hat{\text{cheerios}}$ = 0.56 + 5.11(diameter).


-If the diameter is increased from 4 inches to 14 inches, the predicted number of cheerios will Increase by about……

A) 21
B) 72
C) none of these
D) 51
E) 10

Question 30

The price of first class stamp has followed inflation over time and has increased at a constant percentage over time. The most useful predictive model is probably……

A) quadratic
B) exponential
C) linear
D) power
E) logarithmic

Question 31

The relationship between the number of hours a person practices a task and the time it takes them to complete the task is calculated to have R-sq = 56.7%. The value of the correlation coefficient is

A) 0.753
B) -0.238
C) 0.238
D) -0.753
E) 2.38

Question 32

This regression analysis examines the relationship between the number of years of formal education a person has and their annual income. According to this model, about how much more money do people who finish a 4-year college program earn each year, on average, than those with only a 2-year degree?

Dependent variable is Income
$R$ -squared $=25.8 \%$
$\mathrm{s}=3888$ with 57 degrees of freedom
$\begin{array}{lcc}\text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } \\ \text { Constant } & 3984.45 & 6600 \\ \text { Education } & 2668.45 & 600.1\end{array}$

A) $2710
B) $2006
C) $9321
D) $5337
E) $7968

Question 33

Residuals are...

A) possible models not explored by the researcher.
B) variation in the data that is explained by the model.
C) data collected from individuals that is not consistent with the rest of the group.
D) none of these
E) the difference between observed responses and values predicted by the model.

Question 34

Another company's sales increase by the same percent each year. This growth is...

A) quadratic
B) linear
C) power
D) exponential
E) logarithmic

Question 35

A researcher studied the relationship between family income and amount of money spent on an automobile. She calculated that $\mathrm{R}^{2}$ = 45%. Which is the correct interpretation?

A) None of these
B) 45% of the price of the car can be predicted by using income.
C) The probability of predicting the correct price of a car is 45%.
D) 45% of the variability in car price can be explained by using income.
E) The car price fluctuates 45% more than income.

Question 36

A least squares line of regression has been fitted to a scatterplot; the model's residuals plot is shown. Which is true?

A) The linear model is poor because the correlation is near 0.
B) None of these.
C) The linear model is appropriate.
D) The linear model is poor because some residuals are large.
E) A curved model would be better.

Question 37

A company's sales increase by the same amount each year. This growth is...

A) linear
B) power
C) logarithmic
D) exponential
E) quadratic

Question 38

If a data set has a relationship that is best described by a linear model, than the residual plot will

A) have no pattern with a correlation near 0.
B) also have a linear pattern with a similar correlation.
C) have a curved pattern, like a parabola.
D) be an unknown shape.
E) none of these

Question 39

Which of the following is not a goal of re-expressing data?

A) Make the form of a scatterplot more nearly linear.
B) Make the distribution of a variable more symmetric.
C) Make the scatter in a scatterplot spread out evenly rather than following a fan shape.
D) All of these are goals of re-expressing data.
E) Make the spread of several groups more alike.

Question 40

Computer output in the scenario described in problem #8 reports that s = 2.3. Which is the correct interpretation of this value?

A) The slope of the regression line is $2.30 per light.
B) The initial cost, even with no lights is $2.30.
C) The slope of the regression line is 2.3 lights per dollar.
D) The average prediction error of the regression line is $2.30.
E) The correlation is 2.3.

Question 41

Penicillin assimilation Doctors studying how the human body assimilates medication inject a patient with penicillin, and then monitor the concentration of the drug in the patient's blood for several hours. The data are shown in the table.

$\begin{array}{|c|c|}\hline \text { Time elapsed }&\text { Concentration } \\\text { (Hours) }&\text { (Units/cc) } \\\hline 1 & 42 \\\hline 2 & 28 \\\hline 3 & 19 \\\hline 4 & 13 \\\hline 5 & 9 \\\hline 6 & 6 \\\hline 7 & 4 \\\hline\end{array}$

a. Straighten the scatterplot by re-expressing these data and create an appropriate model for predicting the concentration of penicillin.
b. Use your model to estimate what the concentration of penicillin will be after 8 hours.

Question 42

During a science lab, students heated water, allowed it to cool, and recorded the temperature over time. They computed the difference between the water temperature and the room temperature. The results are in the table.

$\begin{array}{|l|l|l|l|l|l|l|}\hline\text { Time (in minutes) }& 10 & 20 & 30 & 40 & 50 & 60 \\\hline \text { Difference in temp. (degrees F)}& 68 & 36 & 20 & 10 & 6 & 4 \\\hline\end{array}$



-Newton's Law of Cooling suggests an exponential function is appropriate. Re-express the data using logarithms and sketch a new scatterplot.

Question 43

After conducting a marketing study to see what consumers thought about a new tinted contact lens they were developing, an eyewear company reported, "Consumer satisfaction is strongly correlated with eye color." Comment on this observation.

Question 44

An article in the Journal of Statistics Education reported the price of diamonds of different sizes in Singapore dollars
(SGD). The following table contains a data set that is consistent with this data, adjusted to US dollars in 2004:





-Would it be better for a customer buying a diamond to have a negative residual or a positive residual from this model? Explain.

Question 45

A scatterplot of log(Y) vs. log(X) reveals a linear pattern with very little scatter. It is probably true that…

A) the correlation between X and Y is near +1.
B) the residuals plot for regression of Y on X shows a curved pattern.
C) the calculator's LnReg function will model the association between X and Y.
D) the correlation between X and Y is near 0.
E) the scatterplot of Y vs X shows a linear association.

Question 46

Using the equation in number #8 again, if a homeowner doubles the number of lights he uses from 500 to 1000, how much do we predict he will increase his electric bill by?

A) $12.50
B) $10
C) $22.50
D) $2
E) $35

Question 47

Breaking strength A company manufactures polypropylene rope in six different sizes. To assess the strength of the ropes they test two samples of each size to see how much force (in kilograms) the ropes will hold without breaking. The table shows the results of the tests. We want to create a model for predicting the breaking strength from the diameter of the rope.

$\begin{array}{|c|c|c|}\hline \begin{array}{c}\text { Diameter } \\(\mathrm{mm})\end{array} & \begin{array}{c}\text { Strength } \\(\mathbf{k g})\end{array} \\\hline 4 & 60 \mid 76 \\\hline 7 & 157 \mid 153 \\\hline 10 & 254 \mid 262 \\\hline 12 & 334\mid 388 \\\hline 15 & 551 \mid 529 \\\hline 20 & 938 \mid 893 \\\hline\end{array}$

a. Find a model that uses re-expressed data to straighten the scatterplot.
b. The company is thinking of introducing a new 25mm rope. How strong should it be?
(Write a sentence in context based on one of your models.)

Question 48

Maple trees A forester would like to know how big a maple tree might be at age 50 years.
She gathers data from some trees that have been cut down, and plots the diameters (in
inches) of the trees against their ages (in years). First she makes a linear model. The scatterplot and residuals plot are shown.
a. Describe the association shown in the scatterplot.
b. Do you think the linear model is appropriate? Explain.
c. If she uses this model to try to predict the diameter of a 50-year old maple tree, would you expect that estimate to be fairly accurate, too low, or too high? Explain.
Now she re-expresses the data, using the logarithm of age to try to predict the diameter of the tree. Here are the regression analysis and the residuals plot.

$\begin{array}{lcc}\text { Dependent variable is: } & \text { Diam } \\ R \text { squared }=84.3 \% & \\ \text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } \\ \text { Constant } & -8.60770 & 1.681 \\ \text { Log(Age }) & 15.0701 & 1.299\end{array}$


d. Explain why you think this is a better model.
e. Using this model, predict the diameter of a maple tree at age 50 years.

Question 49

A silly psychology student gathers data on the shoe size of 30 of his classmates and their GPA's.
The correlation coefficient between these two variables is most likely to be

A) near 0
B) exactly +1.0
C) near -0.6
D) exactly -1.0
E) near +0.6

Question 50

High Score The longer you play a video game, the higher score you can usually achieve.
An analysis of a popular game found the following relationship between the hours a
player has played a game and their corresponding high score on that game.

Dependent variable is High Score
$\text { R-squared }=76.5 \%$
$\mathrm{s}=383.3$ with 89 degrees of freedom

$\begin{array}{lcc}\text { Variable }&\text { Coefficient}&\text { s.e. of Coeff}\\\text { Constant } & 524.8 & 145.3 \\ \text { Hours } & 2498.8 & 324.5\end{array}$

a. Write the regression equation and define the variables of your equation in context.
b. Interpret the slope in context.
c. Interpret the y-intercept in context.
d. Interpret s in context.
e. What is the correlation coefficient? Interpret this value in context.

Question 51

A study examined the number of trees in a variety of orange groves and the corresponding number of oranges that each grove produces in a given harvest year. Linear regression was calculated and the results are below.

linear regression results:
Dependent Variable:
oranges
Independent Variable: trees
Sample size: 9
R-sq = 0.886 s = 31394.7

$\begin{array}{|l|r|r|}\hline \text { Parameter} & \text {Estimate} & \text { Std. Err. } \\\hline \text { Constant} & 390.59 & 16328.8 \\\hline \text {Trees }& 525.84 & 71.22 \\\hline\end{array}$






-Write the regression equation. Define all variables used in your equation.

Question 52

The following is a scatterplot of the average final exam score versus midterm score for 11 sections of an introductory statistics class:


The correlation coefficient for these data is r = 0.829. If you had a scatterplot of the final exam score versus midterm score for all individual students in this introductory statistics course, would the correlation coefficient be weaker, stronger, or about the same? Explain.

Question 53

During a science lab, students heated water, allowed it to cool, and recorded the temperature over time. They computed the difference between the water temperature and the room temperature. The results are in the table.

$\begin{array}{|l|l|l|l|l|l|l|}\hline \text { Time (in minutes)} & 10 & 20 & 30 & 40 & 50 & 60 \\\hline \text {Difference in temp. (degrees F)} & 68 & 36 & 20 & 10 & 6 & 4 \\\hline\end{array}$



-Use the equation to predict the difference in temperature at time 0 minutes. What does this value represent?

Question 54

Subaru costs Data collected from internet ads for 1999 Subarus were used to create a model to estimate
the asking price of the car based on the number of miles it had been driven. The model has
$\hat{r}$ =0.47
and equation $\hat{\text{ Price}}$ = 15,327 - 0.11(Miles). The plot of residuals versus the predicted price is
shown.
a. Do you think you could make accurate estimates of Subaru prices with this model? Explain.
b. Interpret the slope of the line.
c. One of the cars in the data set had been driven 42,000 miles. How much was the owner
asking for it? (Show work.)

Question 55

Identify what is wrong with each of the following statements:
a. The correlation between Olympic gold medal times for the 800m hurdles and year is -0.66 seconds per year.
b. The correlation between Olympic gold medal times for the 100m dash and year is -1.37. c. Since the correlation between Olympic gold medal times for the 800m hurdles and 100m dash is -0.41, the correlation between times for the 100m dash and the 800m hurdles is +0.41.
d. If we were to measure Olympic gold medal times for the 800m hurdles in minutes instead of seconds, the correlation would be -0.66/60 = -0.011.

Question 56

If the point in the upper left corner of the scatterplot is removed, what will happen to the correlation (r) and the slope of the line of best fit (b)?

A) r will decrease and b will increase.
B) Both will decrease.
C) r will increase and b will decrease.
D) They will not change.
E) Both will increase.

Question 57

Current research states that a good diet should contain 20-35 grams of dietary fiber. Research also states that each day should start with a healthy breakfast. The nutritional information for 77 breakfast cereals was reviewed to find the grams of fiber and the number of calories per serving. The scatterplot below shows the relationship between fiber and calories for the cereals.



-Do you think there is a clear pattern? Describe the association between fiber and calories.

Question 58

Baseball coaches use a radar gun to measure the speed of pitcher’s fastball. They also record outcomes such as hits and strikeouts. The scatterplot below shows the relationship between the average speed of a fastball and the average number of strikeouts per nine innings for each pitcher on the Bulldogs, based on the past season.




-Do you think the association would be stronger or weaker if we used data from one month of the season?

Question 59

If you have a puppy who you are feeding Hill's Science Diet® original puppy food, the feeding guidelines for puppies who are 10 to 12 months old are as follows (Source: Hill's feeding guidelines):
$\begin{array}{l|cccccccccc} \text {Weight (lbs) }& 2 & 5 & 8 & 10 & 20 & 40 & 60 & 80 & 100 & 120 \\\hline\text { Food (cups)} & 1 / 3 & 2 / 3 & 2 & 1-1 / 8 & 2 & 3-1 / 4& 4-1 / 3 & 5.5 & 6.5 & 7-1 / 3 \end{array}$

A scatterplot of the data is:
Does it seem reasonable to perform a linear regression to predict amount of food from the puppy's weight based on this data set? Explain.

Question 60

After conducting a survey at a pet store to see what impact having a pet had on the condition of the yard, a news reporter stated "There appears to be a strong correlation between owning a pet and the condition of the yard." Comment on this observation.

Question 61

Car commercials A car dealer investigated the association between the number of TV commercials he ran each week and the number of cars he sold the following weekend. He found the correlation to be r = 0.56. During the time he collected the data he ran an average of 12.4 commercials a week with a standard deviation of 1.8, and sold an average
of 30.5 cars with a standard deviation of 4.2. Next weekend he is planning a sale, hoping to sell 40 cars. Create a linear model to estimate the number of commercials he should run
this week. Write a sentence explaining your recommendation.

Question 62

A school board study found a moderately strong negative association between the number of hours high school seniors worked at part-time jobs after school hours and the students' grade point averages.
a. Explain in this context what "negative association" means.
b. Hoping to improve student performance, the school board passed a resolution urging
parents to limit the number of hours students be allowed to work. Do you agree or disagree with the school board's reasoning? Explain.

Question 63

A study examined the number of trees in a variety of orange groves and the corresponding number of oranges that each grove produces in a given harvest year. Linear regression was calculated and the results are below.
linear regression results:
Dependent Variable: oranges
Independent Variable: trees
Sample size: 9
R-sq = 0.886 s = 31394.7

$\begin{array}{|l|r|r|}\hline\text { Parameter} & \text {Estimate} &\text { Std. Err. } \\\hline \text {Constant }& 390.59 & 16328.8 \\\hline \text {Trees} & 525.84 & 71.22 \\\hline\end{array}$




-Interpret s in context.

Question 64

An article in the Journal of Statistics Education reported the price of diamonds of different sizes in Singapore dollars (SGD). The following table contains a data set that is consistent with this data, adjusted to US dollars in 2004:




-Create a model to predict diamond costs from the size of the diamond.

Question 65

A study examined the number of trees in a variety of orange groves and the corresponding number of oranges that each
grove produces in a given harvest year. Linear regression was calculated and the results are below.
linear regression results:
Dependent Variable: oranges
Independent Variable: trees
Sample size: 9
R-sq = 0.886
s = 31394.7

- $\begin{array}{|l|r|r|}\hline \text { Parameter} & \text { Estimate }& \text { Std. Err. } \\\hline \text { Constant }& 390.59 & 16328.8 \\\hline \text { Trees }& 525.84 & 71.22 \\\hline\end{array}$

Since $r^{2}$ is not 100%, there must be other factors in influencing the number of oranges harvested. What percentage is that and what is another factor you think might be involved?

Question 66

Math and Verbal Suppose the correlation between SAT Verbal scores and Math scores is0.57 and that these scores are normally distributed. If a student's Verbal score places her atthe 90th percentile, at what percentile would you predict her Math score to be? (Show your work.)

Question 67

An article in the Journal of Statistics Education reported the price of diamonds of different sizes in Singapore dollars
(SGD). The following table contains a data set that is consistent with this data, adjusted to US dollars in 2004:



-Interpret the intercept of your model in context.

Question 68

Baseball coaches use a radar gun to measure the speed of pitcher’s fastball. They also record outcomes such as hits and strikeouts. The scatterplot below shows the relationship between the average speed of a fastball and the average number of strikeouts per nine innings for each pitcher on the Bulldogs, based on the past season.




-Do you think there is a pattern? Describe the association between speed and the number of strikeouts.

Question 69

Time Wasted A group of students decide to see if there is link between wasting time on the internet and GPA. They don't expect to find an extremely strong association, but they're hoping for at least a weak relationship. Here are the findings.

$\begin{array}{|l|l|l|}\hline \text { linear regression results: }& & \\ \text { Dependent Variable: GPA} & & \\ \text { Sample size: 10 }& & \\ \text { R (correlation coefficient) }=-0.37199274 & & \\ \text { R-sq =0.1383786 } & & \\\mathrm{~s}=0.85365134 & & \\& & \\ \text { Parameter }& \text { Estimate} & \text { Std. Err.} \\\hline \text { Intercep}t & 4.06191 & 0.74405 \\\hline \text { Hours/week} & -0.0297 & 0.02616 \\\hline\end{array}$

a. How strong is the relationship the students found? Describe in context with statistical justification.
One student is concerned that the relationship is so weak, there may not actually be any relationship at all. To test this concern, he runs a simulation where the 10 GPA's are randomly matched with the 10 hours/week. After each random assignment, the correlation is calculated. This process is repeated 100 times. Here is a histogram of the 100 correlations. The correlation coefficient of -0.371 is indicated with a vertical line.
b. Do the results of this simulation confirm the suspicion that there may not be any relationship? Refer specifically to the graph in your explanation.

Question 70

Baseball coaches use a radar gun to measure the speed of pitcher’s fastball. They also record outcomes such as hits and strikeouts. The scatterplot below shows the relationship between the average speed of a fastball and the average number of strikeouts per nine innings for each pitcher on the Bulldogs, based on the past season.



-Comment on any unusual data point or points in the data set. Explain.

Question 71

The bigger the stop sign, the more expensive it is. Here is a graph of the height of a sign in inches versus its cost in dollars.




-Describe why performing linear regression with these data is not a good decision.

Question 72

Baseball coaches use a radar gun to measure the speed of pitcher’s fastball. They also record outcomes such as hits and strikeouts. The scatterplot below shows the relationship between the average speed of a fastball and the average number of strikeouts per nine innings for each pitcher on the Bulldogs, based on the past season.




-Do you think a model based on these data could accurately predict the average number of strikeouts for a pitcher with an average fastball speed of 70 mph.? Explain.

Question 73

An article in the Journal of Statistics Education reported the price of diamonds of different sizes in Singapore dollars
(SGD). The following table contains a data set that is consistent with this data, adjusted to US dollars in 2004:



-Do you think a linear model is appropriate here? Explain.

Question 74

Crawling Researchers at the University of Denver Infant Study Center investigated whether babies take longer to learn to crawl in cold months (when they are often bundled in clothes that restrict their movement) than in warmer months. The study sought an association between babies' first crawling age (in weeks) and the average temperature during the month they first try to crawl (about 6 months after birth). Between 1988 and
1991 parents reported the birth month and age at which their child was first able to creep
or crawl a distance of four feet in one minute. Data were collected on 208 boys and 206 girls. The graph below plots average crawling ages (in weeks) against the mean temperatures when the babies were 6 months old. The researchers found a correlation of r
= -0.70 and the their line of best fit was

$\hat{\text{AvAge}}$ = 36 - 0.08AvTemp.


a. Draw the line of best fit on the graph. (Show your method clearly.)
b. Describe the association in context.
c. Explain (in context) what the slope of the line means.
d. Explain (in context) what the y-intercept of the line means. e. Explain (in context) what $R^{2}$ means.
f. In this context, what does a negative residual indicate?

Question 75

Music and grades (True Story) A couple of years ago a local newspaper published research results claiming a positive association between the number of years high school children had taken instrumental music lessons and their performances in school (GPA).
a. What does "positive association" mean in this context?
b. A group of parents then went to the School Board demanding more funding for music
programs as a way to improve student chances for academic success in high school. As a statistician, do you agree or disagree with their reasoning? Explain briefly.

Question 76

Gas mileage An important factor in the amount of gasoline a car uses is the size of the engine. Called "displacement", engine size measures the volume of the cylinders in cubic inches. The regression analysis is shown.

Dependent variable is: $\quad$ $\quad$ MPG
89 total cases of which 0 are missing
$\mathrm{R}$ squared $=60.9 \% \quad \mathrm{R}$ squared (adjusted) $=60.0 \%$
$s=3.056$ with $89-2=87$ degrees of freedom
$\begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \\ \text { Regression } & 696.744 & 1 & 696.744 & 74.6 \\ \text { Residual } & 448.236 & 48 & 9.33826 & \\ & & & & \\ \text { Variable } & \text { Coefficient } & \text { s.e. of Coeff } & \text { t-ratio } & \text { prob } \\ \text { Constant } & 34.9799 & 1.231 & 28.4 & <0.0001 \\ \text { Eng. Displcmt } & -0.066196 & 0.0077 & -8.64 & <0.0001\end{array}$

a. How many cars were included in this analysis?
b. What is the correlation between engine size and fuel economy?
c. A car you are thinking of buying is available with two different size engines, 190 cubic inches or 240 cubic inches. How much difference might this make in your gas mileage? (Show your work.)

Question 77

Current research states that a good diet should contain 20-35 grams of dietary fiber. Research also states that each day should start with a healthy breakfast. The nutritional information for 77 breakfast cereals was reviewed to find the grams of fiber and the number of calories per serving. The scatterplot below shows the relationship between fiber and calories for the cereals.



-Do you think a model could accurately predict the number of calories in a serving of cereal that has 22 grams of fiber? Explain.

Question 78

A study examined the number of trees in a variety of orange groves and the corresponding number of oranges that each grove produces in a given harvest year. Linear regression was calculated and the results are below.

linear regression results:
Dependent Variable: oranges
Independent Variable: trees
Sample size: 9
R-sq = 0.886 s = 31394.7

$\begin{array}{|l|r|r|}\hline \text {Parameter} & \text { Estimate} & \text {Std. Err. } \\\hline \text {Constant }& 390.59 & 16328.8 \\\hline \text {Trees} & 525.84 & 71.22 \\\hline\end{array}$





-Does the value of s concern you? How might you deal with this data differently to address this problem?

Question 79

Earning power A college's job placement office collected data about students' GPAs and the salaries they earned in their first jobs after graduation. The mean GPA was 2.9 with a standard deviation of 0.4. Starting salaries had a mean of $47,200 with a SD of $8500. The correlation between the two variables was r = 0.72. The association appeared to be linear in the scatterplot. (Show work.)
a. Write an equation of the model that can predict salary based on GPA. b. Do you think these predictions will be reliable? Explain.
c. Your brother just graduated from that college with a GPA of 3.30. He tells you that based on this model the residual for his pay is -$1880. What salary is he earning?

Question 80

An article in the Journal of Statistics Education reported the price of diamonds of different sizes in Singapore dollars
(SGD). The following table contains a data set that is consistent with this data, adjusted to US dollars in 2004:




-Make a scatterplot and describe the association between the size of the diamond (carat) and the cost (in US dollars).