AP Statistics - TVD: Two-Variable Data

1 Exploratory Analysis > Two-variable data analysis · Level 3

Large student debt is a growing problem for college graduates, with many graduates seemingly unable ever to get out of debt. Suppose the average college debt for graduates is \$120,000 with a standard deviation of $40000, and suppose the average stress level of college graduates on a 1–100 scale is 30 with a standard deviation of 10. If the correlation between the stress level and college debt is 0.6, w hat is the least squares linear regression equation for stress level among college graduates based on their amount of debt (in \$1,000)?

A

Predicted stress level = 12 + 0.15(College debt in \$1,000)

B

Predicted stress level = 113.5 + 0.15(College debt in \$1,000)

C

Predicted stress level = 30 + 2.4(College debt in \$1,000)

D

Predicted stress level = 0.25(College debt in \$1,000)

E

Predicted stress level = 28.2 + 0.015(College debt in \$1,000)

2 Exploratory Analysis > Two-variable data analysis · Level 3

Suppose every student who attended Saturday review classes scored exactly 10 points higher on a final exam than he/she did on a midterm exam. What would be the correlation between scores on the final exam and scores on the midterm exam for students attending the review classes?

A

0

B

Somewhat positive

C

0.5

D

Nearly 1

E

Exactly 1

3 Exploratory Analysis > Two-variable data analysis · Level 3

Paper lengths and final grades are obtained from a random sample of term papers handed in to a particular high school English teacher. The resulting regression equation is shown. Grade = 52.34 + 1.24(Length) with $r = 0.18$ What percentage of the variation in final grades can be explained by the linear regression model of Grade on Length?

A

0.24 percent

B

1.24 percent

C

3.24 percent

D

18 percent

E

24 percent

4 Exploratory Analysis > Two-variable data analysis · Level 3

In a study of winning percentages in home games versus average home attendance for Big Ten football teams, the resulting regression line is shown in the following equation. $\text{Winning\%} = 39 + 0.00025(\text{Attendance})$ What is the residual if a team has a winning percentage of 55% with an average attendance of 75,000?

A

–3.025

B

–2.75

C

2.75

D

3.025

E

57.75

5 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following is an incorrect statement about the least square regression line?

A

It always passes through the point $(\overline{x}, \overline{y})$.

B

The slope always has the same sign as the correlation.

C

It minimizes the sum of the squares of the residuals.

D

The slope is always between –1 and +1.

E

The correlation of y versus x is the same as the correlation of x versus y.

6 Exploratory Analysis > Two-variable data analysis · Level 3

A linear regression analysis relating teachers' salaries to years of experience yields: $\hat{y} = 37.15 + 2.405x$ where x is years of experience and y is salary (in \$1,000). Which of the following is the most proper conclusion?

A

A starting teacher will earn \$37,150, while one with 70 years of experience should earn \$205,500.

B

Starting teachers average \$37,150 with bonuses of \$2,405 every year.

C

There is a cause-and-effect relationship between teachers' salaries and experience with each extra year of experience corresponding to an extra \$2,405 in salary.

D

Starting salaries for teachers average \$37,150, and each year of experience is associated with an average increase of \$2,405.

E

There is a high correlation between teachers' salaries and years of experience.

7 Exploratory Analysis > Two-variable data analysis · Level 3

What is the correct regression output for the scatterplot below?

A

B

C

D

E

8 Exploratory Analysis > Two-variable data analysis · Level 3

Suppose a correlation is positive. Given two points from the scatterplot, which of the following is/are possible? I. The first point has a smaller x-value and a smaller y-value than the second point. II. The first point has a larger x-value and a larger y-value than the second point. III. The first point has a smaller x-value and a larger y-value than the second point.

A

I only

B

II only

C

III only

D

I and II only

E

I, II, and III

9 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following statements about the correlation coefficient r is incorrect?

A

It is not affected by changes in the measurement units of the variables.

B

It itself never has units.

C

It gives information about a linear association, not about causation.

D

It always takes values between –1 and +1 unless the association is nonlinear.

E

It is not affected by which variable is explanatory and which is response.

10 Exploratory Analysis > Two-variable data analysis · Level 3

The value of a new luxury automobile (in \$1,000) versus age (in years) has the following regression analysis. What is the meaning of –4.78869 in the computer output?

A

The value of the automobile will decrease by 4.78869 thousand dollars for every increase of 1 year in the age of the automobile.

B

On average, there is a predicted decrease in the value of \$1,000 for every increase of 4.78869 years in the age of the automobile.

C

On average, there is a predicted decrease of 4.78869 thousand dollars in the value for every increase of 1 year in the age of the automobile.

D

Approximately 4.79 percent of the variability in value can be explained by the linear model of value versus age.

E

The standard deviation of the residuals is approximately 4.79.

11 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the scatterplots below could have resulted in the residual plot shown above? (The y-scales are not the same in the scatterplots as in the residual plot.)

A

B

C

D

E

None of these could result in the given residual plot.

12 Exploratory Analysis > Two-variable data analysis · Level 3

Consider the scatterplot below of starting salaries and salaries after 5 years of employment for 20 executives at a software company. Which of the statements below is incorrect?

A

Both the correlation and the slope are positive.

B

More executives started making under \$50,000 than were making under \$50,000 after five years.

C

The same number of executives started at \$100,000 as were making \$100,000 after five years.

D

Most executives who started making under \$50,000 were still making under \$50,000 after five years.

E

Some executives were making less after five years than they made initially.

13 Exploratory Analysis > Two-variable data analysis · Level 3

Consider the following three scatterplots. Which scatterplot has the greatest correlation coefficient $r$?

A

I

B

II

C

III

D

They all have the same correlation coefficient.

E

The question cannot be answered without additional information.

14 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following statements about influential points is incorrect?

A

Removal of an influential point sharply affects the regression line.

B

Looking at a residual plot is a helpful way of identifying influential points.

C

Determining a regression model with and without a point is an excellent way of identifying influential points.

D

Outliers in the $x$-direction are more likely to be influential points than are outliers in the $y$-direction.

E

None of the above are incorrect statements.

15 Exploratory Analysis > Two-variable data analysis · Level 3

A regression analysis on the relationship between average annual profits of financial companies (in millions of dollars) and the yearly number of major security breaches gives the following computer output. If the analysis was rerun using the number of breaches as the dependent variable instead of profits, what would be the correlation coefficient?

A

–0.318

B

0.465

C

–0.465

D

0.682

E

–0.682

16 Exploratory Analysis > Two-variable data analysis · Level 3

Suppose the regression line for a set of data, $\hat{y} = a + 5x$, passes through the point (1, 7). If $\overline{x}$ and $\overline{y}$ are the sample means of the $x$- and $y$-values, respectively, what is $\overline{y}$?

A

$\overline{x}$

B

$5\overline{x}$

C

$6 + \overline{x}$

D

$2 + 5\overline{x}$

E

$7 + 5\overline{x}$

17 Exploratory Analysis > Two-variable data analysis · Level 3

The number of students who were victims of bullying at a high school during the years 2010–2017 is fitted with a least squares regression line. The graph of the residuals and some computer output is as follows. How many students were victims of bullying at this high school in the year 2013?

A

47

B

48

C

52

D

53

E

58

18 Exploratory Analysis > Two-variable data analysis · Level 3

A simple random sample of 25 AP Statistics students provides the following statistics. Number of hours studying for this class each day: $\overline{x} = 1.8$, $s_x = 0.5$ Average grade for this class: $\overline{y} = 86.5$, $s_y = 4.2$ Correlation $r = 0.45$ Based on this data, what is the resulting linear regression equation?

A

$\hat{\text{Grade}} = 52.9 + 18.67(\text{Hours})$

B

$\hat{\text{Grade}} = 79.7 + 3.78(\text{Hours})$

C

$\hat{\text{Grade}} = 83.1 + 1.89(\text{Hours})$

D

$\hat{\text{Grade}} = 85.7 + 0.45(\text{Hours})$

E

$\hat{\text{Grade}} = 86.4 + 0.0536(\text{Hours})$

19 Exploratory Analysis > Two-variable data analysis · Level 3

A biologist studying robins collected data on the weight and the egg production for 50 female birds. The correlation is found to be 0.7. Which of the following is a true statement?

A

On average, a 70 percent increase in weight results in a 49 percent increase in egg production.

B

On average, a 70 percent increase in weight results in a 100 percent increase in egg production.

C

Seventy percent of a robin's egg production can be explained by the bird's weight.

D

Seventy percent of the variation in robin egg production can be accounted for by the linear regression model of egg production on bird weight.

E

Greater egg production tends to be associated with greater bird weight.

20 Exploratory Analysis > Two-variable data analysis · Level 3

One model for global warming is given by $\hat{y} = 315 + 1.52x$ with $r = 0.92$, where $y$ is CO₂ atmospheric concentration in ppm and $x$ is years since 1960. What is the correct interpretation of the slope?

A

On average, atmospheric CO₂ concentration has been rising by 1.52 ppm per year since 1960.

B

The baseline atmospheric CO₂ concentration is 315 ppm.

C

The regression model explains 84.64 percent of the variation of atmospheric CO₂ concentration over the years since 1960.

D

The regression model explains 92 percent of the variation of atmospheric CO₂ concentration over the years since 1960.

E

Atmospheric CO₂ concentration will top 500 ppm in the year 2082.

21 Exploratory Analysis > Two-variable data analysis · Level 3

Consider the points (–2, 3), (0, 7), (1, 9), (3, 12), and (10, $n$). What should $n$ be so that the correlation between the $x$ and $y$ values is $r = 1$?

A

25

B

26

C

27

D

A value different from any of the above

E

No value for $n$ can make $r = 1$.

22 Exploratory Analysis > Two-variable data analysis · Level 3

Suppose the data from two scatterplots, A and B, result in identical least squares regression lines with positive slopes. Which of the following statements is true?

A

The correlation in A equals the correlation in B.

B

The sum of the squares of the residuals in A equals the sum of the squares of the residuals in B.

C

If the sum of the squares of the residuals in A is greater than the sum of the squares of the residuals in B, then the correlation in A will be greater than the correlation in B.

D

If the sum of the squares of the residuals in A is greater than the sum of the squares of the residuals in B, then the correlation in A will be less than the correlation in B.

E

None of the above are true statements.

23 Exploratory Analysis > Two-variable data analysis · Level 3

Two drivers, one for Uber and one for Lyft, compare their weekly miles driven for 17 consecutive weeks. The following scatterplot is the result, where the units for both axes are miles. Which statement below gives the best comparison between the weekly miles driven by the two drivers?

A

In all 17 weeks, the Lyft driver drove more miles than the Uber driver.

B

In all 17 weeks, the Uber driver drove more miles than the Lyft driver.

C

In all but 1 week, the Lyft driver drove more miles than the Uber driver.

D

In all but 1 week, the Uber driver drove more miles than the Lyft driver.

E

In all but 3 weeks, the Lyft driver drove more miles than the Uber driver.

24 Exploratory Analysis > Two-variable data analysis · Level 3

Suppose a study finds that the correlation coefficient relating outside temperature in degrees Fahrenheit during the winter to the amount of natural gas a house consumes in cubic feet per day is $r = -1$. Which of the following is a proper conclusion?

A

Lower outside winter temperatures cause a greater usage of natural gas.

B

Higher outside winter temperatures cause a greater usage of natural gas.

C

There is a 100% cause-and-effect relationship between outside winter temperatures and natural gas usage.

D

There is a very strong association between outside winter temperatures and natural gas usage.

E

None of the above are proper conclusions.

25 Exploratory Analysis > Two-variable data analysis · Level 3

What is the correct scatterplot for the computer output shown above?

A

B

C

D

E

26 Exploratory Analysis > Two-variable data analysis · Level 3

Suppose the correlation between two variables is –0.28. If each of the y-values is multiplied by –2, which of the following is true about the new scatterplot?

A

It slopes up to the right, and the correlation is –0.28.

B

It slopes up to the right, and the correlation is +0.28.

C

It slopes down to the right, and the correlation is –0.28.

D

It slopes down to the right, and the correlation is –0.56.

E

It slopes up to the right, and the correlation is +0.56.

27 Exploratory Analysis > Two-variable data analysis · Level 3

Consider $n$ pairs of numbers. Suppose $\overline{x} = 3$, $s_x = 2$, $\overline{y} = 5$, and $s_y = 4$. Which of the following could be the least squares regression line?

A

$\hat{y} = 11 - 2x$

B

$\hat{y} = -4 + 3x$

C

$\hat{y} = 17 - 4x$

D

$\hat{y} = -10 + 5x$

E

$\hat{y} = 23 - 6x$

28 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following are possible residual plots?

A

I only

B

II only

C

III only

D

I and II only

E

I, II, and III

29 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following is a true statement about the correlation coefficient $r$?

A

A correlation of 0.4 means that 40 percent of the points are highly correlated.

B

The unit of measurement for correlation is the y-unit per x-unit.

C

Multiplying every y-value by –1 leaves the correlation unchanged.

D

Perfect correlation, that is, when the points lie exactly on a straight line, results in $r = 0$.

E

The square of the correlation measures the proportion of the y- variance that is predictable from the linear regression model.

30 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following statements about correlation $r$ is incorrect?

A

The correlation and the slope of the regression line always have the same sign.

B

Correlation $r$ measures the strength and direction of only linear association.

C

Outliers can greatly affect the value of $r$.

D

A correlation of –0.48 and a correlation of +0.48 show the same degree of clustering around the regression line.

E

A correlation of 0.48 indicates a relationship that is three times as linear as one for which the correlation is 0.16.

31 Exploratory Analysis > Two-variable data analysis · Level 3

A scatterplot of a town's population versus time indicates a possible exponential relationship. A linear regression on $y = \log(\text{population \in 1,000s})$ against $x = \text{years since 2015}$ gives $\hat{y} = 2.4 + 0.03x$ with $r = 0.72$. Which of the following is a valid conclusion?

A

On average, population goes up 0.03 thousand per year.

B

The predicted population for year 2025 is approximately 501 thousand.

C

51.84 percent of the variation in population can be explained by variation in time.

D

72 percent of the variation in population can be explained by variation in time.

E

None of the above are valid conclusions.

32 Exploratory Analysis > Two-variable data analysis · Level 3

A scatterplot of log X and log Y shows a strong negative correlation close to –1. Which of the following is a true statement?

A

The variables X and Y also have a correlation close to –1.

B

A scatterplot of the variables X and Y shows a nonlinear pattern.

C

The residual plot of the variables X and Y shows a random pattern.

D

The residual plot of the variables log X and log Y shows a strong nonrandom pattern.

E

None of the above are true.

33 Exploratory Analysis > Two-variable data analysis · Level 3

A random sample of 12 cigarettes of different brands is selected. The tar and nicotine content of each is measured and graphed, resulting in the scatterplot below. If the point labeled X is removed, which of the following statements would be true about the least squares regression line and the correlation coefficient?

A

Both the slope and the correlation would remain the same.

B

Both the slope and the correlation would increase.

C

Both the slope and the correlation would decrease.

D

The slope would increase, and the correlation would decrease.

E

The slope would decrease, and the correlation would increase.

34 Exploratory Analysis > Two-variable data analysis · Level 3

Ten overweight people went on a new diet, and each lost exactly 7 pounds. What is the correlation between weights before the diet and weights after the diet?

A

0

B

0.5

C

1

D

This cannot be answered without seeing the actual data.

E

This cannot be answered without knowing if the people were randomly picked from the population of all overweight people.

35 Exploratory Analysis > Two-variable data analysis · Level 3

A scatterplot with one point labeled X is shown below. With regard to the least square regression line, which of the following statements is true?

A

X is an influential point.

B

X is an outlier.

C

X has the largest residual, in absolute value, of any point on the scatterplot.

D

There will be no pattern in the residual plot.

E

None of the above are true statements.

36 Exploratory Analysis > Two-variable data analysis · Level 3

Consider the following three scatterplots. Which of the following is true about the correlations for the three scatterplots?

A

None are close to 0.

B

One is approximately 0, one is negative, and one is positive.

C

One is approximately 0, and both of the others are positive.

D

Two are approximately 0, and the other is +1.

E

Two are approximately 0, and the other is close to +1.

37 Exploratory Analysis > Two-variable data analysis · Level 3

In a random sample of eight fast food hamburgers, the association between calories and fat is summarized in the computer output below. Which of the following is the best approximation of the actual calories for the hamburger in the sample that had 15 grams of fat?

A

410

B

415

C

420

D

425

E

430

38 Exploratory Analysis > Two-variable data analysis · Level 3

A recent study on distance (in feet) drivers can see versus age (in years) of drivers resulted in a linear regression model: Predicted distance a driver can see = 576.7 – 3.01(Age) Which of the following is the best interpretation of the slope in context?

A

For each year a driver's age increases, he or she sees 3.01 feet less.

B

For each 3.01 years a driver's age increases, he or she sees 1 foot less.

C

For each year a driver's age increases, he or she sees 3.01 feet less up to a distance of 576.7 feet.

D

For each 3.01 years a driver's age increases, he or she sees 1 foot less on average.

E

For each year a driver's age increases, he or she sees 3.01 feet less on average.

39 Exploratory Analysis > Two-variable data analysis · Level 3

Given the following three scatterplots with correlations $r_1$, $r_2$, and $r_3$, respectively, what is the proper ordering of their correlations?

A

$r_3 < r_1 < r_2$

B

$r_2 < r_1 < r_3$

C

$r_2 < r_3 < r_1 < |r_2|$

D

$r_1 < r_3 < |r_2|$

E

$r_3 < r_2 < r_1 < |r_2|$

40 Exploratory Analysis > Two-variable data analysis · Level 3

Consider the three points (3, 50), (4, 38), and (5, 14). Given any straight line, we can calculate the sum of the squares of the three vertical distances from these points to the line. What is the smallest possible value of this sum?

A

4.9

B

8

C

9.8

D

24

E

40

41 Exploratory Analysis > Two-variable data analysis · Level 3

Four pairs of data are used in determining the regression line $\hat{y} = 8 + 2x$. If the four values of the independent variable are 17, 29, 35, and 43, what is the mean of the four values of the dependent variable?

A

31

B

62

C

70

D

256

E

The mean cannot be determined from the given information.

42 Exploratory Analysis > Two-variable data analysis · Level 3

Suppose the correlation between two variables is $r = 0.16$. What will the new correlation be if every value of the x-variable is tripled, 0.04 is added to all values of the y-variable, and the two variables are then interchanged?

A

0.16

B

0.20

C

0.48

D

0.52

E

0.60

43 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following statements about the correlation $r$ is true?

A

When $r = 0$, there is no relationship between the variables.

B

When $r = 0.3$, 30 percent of the variables are closely related.

C

When $r = 1$, there is a perfect cause-and-effect relationship between the variables.

D

A correlation close to 1 means that a linear model will give the best fit to the data.

E

All of the above statements are false.

44 Exploratory Analysis > Two-variable data analysis · Level 3

Which of the following statements about residuals in a linear regression model is incorrect?

A

The mean of the residuals is always zero.

B

The sum of the residuals is always zero.

C

The regression line for a residual plot is a horizontal line.

D

The standard deviation of the residuals gives a measure of how the points in the scatterplot are spread around the regression line.

E

A residual equals the predicted y minus the observed y.

45 Exploratory Analysis > Two-variable data analysis · Level 3

Data on ages (in months) and eBay prices for used cellphones result in a regression line. $\text{Price} = 500 - 22.5(\text{Age})$ Given that 56.25% of the variation in price is explained by the linear regression model of Price on Age, what is the value of the correlation coefficient $r$?

A

-0.75

B

-0.5625

C

0.5625

D

0.75

E

There is insufficient information to answer this question.

46 Exploratory Analysis > Two-variable data analysis · Level 3

Studies have shown a strong, positive, linear relationship between temperature and the frequency of a cricket's chirps. What does "strong" mean in this context?

A

The temperature is related to the frequency of a cricket's chirps.

B

For each unit increase in the frequency of cricket chirps, the predicted temperature changes by a constant amount.

C

The greater the frequency of cricket chirps, the greater the temperature, on average.

D

The actual temperatures are very close to the temperatures predicted by the linear model.

E

The variability in temperatures is related to the variability in the frequency of cricket chirps.

47 Exploratory Analysis > Two-variable data analysis · Level 3

A study of family incomes and SAT scores reports a correlation of $r = +1.07$. What should be concluded?

A

Students coming from higher-income families tend to have higher SAT scores than students from lower-income families.

B

Students coming from higher-income families have 7 percent higher SAT scores on average than students coming from lower-income families.

C

There is less than a 10 percent relationship between family incomes and SAT scores.

D

There is a strong positive association between family income and SAT scores, but concluding causation would be wrong.

E

A mistake in arithmetic was made.

48 Exploratory Analysis > Two-variable data analysis · Level 3

An HR officer at a high-pressure company surveys executives as to their happiness level and accesses their salaries. The data are displayed in the scatterplot below. Describe the relationship between happiness level and salary.

A

Negative, weak, linear

B

Negative, moderate, linear

C

Negative, strong, nonlinear

D

Positive, strong, nonlinear

E

Positive, weak, nonlinear

49 Exploratory Analysis > Two-variable data analysis · Level 3

Data on $w$ = weight of a car (in pounds) and $s$ = stopping distance (in feet) when the car is traveling at 50 mph and brakes are applied results in the regression equation $\hat{s} = 35 + 0.03w$. Which of the following is the best interpretation of the intercept 35 in the equation?

A

For each 1 pound increase in car weight, the stopping distance increases by 35 feet, on average.

B

For each 1 foot increase in stopping distance, the car weight is 35 pounds greater, on average.

C

The stopping distance is 35 feet when the car weight is 0, on average.

D

The stopping distance is 0 feet when the car weight is 35 pounds, on average.

E

The intercept, 35 feet, has no reasonable interpretation because a weight of 0 is outside the domain of car weights.

50 Exploratory Analysis > Two-variable data analysis · Level 3

A linear regression analysis of rushing yards and points scored in a random sample of 25 NFL games results in the following equation. $\text{Points scored} = 10 + 0.05(\text{Rushing yards})$ A coach would like to predict rushing yards from points scored. He solves the above equation and comes up with the following. $\text{Rushing yards} = -200 + 20(\text{Points scored})$ Did the coach calculate correctly?

A

Yes, because if $y = 10 + 0.05x$, solving algebraically for $x$ gives $x = -200 + 20y$.

B

Yes, because the x- and y-variables are interchangeable in linear regression.

C

Yes, because $\dfrac{1}{0.05} = 20$ showing that the slopes are reciprocals.

D

No, because the equations have different sets of variables.

E

This cannot be answered without knowing the value of the correlation $r$.

51 Exploratory Analysis > Two-variable data analysis · Level 3

A study of airfare prices versus distances gives rise to the following scatterplot. A least squares regression analysis is performed. Which of the following would be apparent from a residual plot of residuals versus distance?

A

The variation in airfare is different across the distances.

B

There is a positive linear relationship between the residuals and distance.

C

The sum of the residuals is less than 0.

D

The sum of the residuals is greater than 0.

E

The residuals go approximately from 100 to 400.

52 Exploratory Analysis > Two-variable data analysis · Level 3

In a random sample of 300 men and 200 women, 50 of the men and 40 of the women said that they brush their teeth from side to side rather than up and down. If there is no difference between the proportions of men and women who brush their teeth from side to side rather than up and down, how many men and women in the sample would be expected to brush their teeth from side to side rather than up and down?

A

45 men and 45 women

B

50 men and 40 women

C

54 men and 36 women

D

150 men and 100 women

E

250 men and 160 women

53 Exploratory Analysis > Two-variable data analysis · Level 3

A random sample of 200 people was anonymously surveyed as to whether they sneak food into movie theaters rather than pay the high price of snacks. 120 were teenagers, and 80 were adult. 150 answered "yes," they have snuck in food, and 50 answered "no." Among those surveyed, there was independence between teenager/adult and whether or not they snuck food into movie theaters. Which of the following tables shows this conclusion?

A

	Yes	No	Total		-----------	-----	-----	-------		Teenagers	75	45	120		Adults	75	5	80		Total	150	50	200

B

	Yes	No	Total		-----------	-----	-----	-------		Teenagers	80	40	120		Adults	70	10	80		Total	150	50	200

C

	Yes	No	Total		-----------	-----	-----	-------		Teenagers	90	30	120		Adults	60	20	80		Total	150	50	200

D

	Yes	No	Total		-----------	-----	-----	-------		Teenagers	110	10	120		Adults	40	40	80		Total	150	50	200

E

	Yes	No	Total		-----------	-----	-----	-------		Teenagers	100	20	120		Adults	50	30	80		Total	150	50	200

54 Exploratory Analysis > Two-variable data analysis · Level 3

In the following table, what value of $n$ results in a table showing perfect independence?

A

30

B

50

C

70

D

75

E

100

55 Exploratory Analysis > Two-variable data analysis · Level 3

A random sample of 200 movies were cross-classified by genre and rating. The following table gives the individual counts.

	G	PG	PG-13	R	Total
Action	5	20	15	5	45
Comedy	10	25	10	5	50
Documentary	25	10	3	2	40
Drama	20	15	17	13	65
Total	60	70	45	25	200

Of the movies in the sample that were classified as PG or PG-13, what proportion of them were either action or drama?

A

$\dfrac{45+65}{200}$

B

$\dfrac{70+45}{200}$

C

$\dfrac{20+15}{70+45}$

D

$\dfrac{20+15+15+17}{200}$

E

$\dfrac{20+15+15+17}{70+45}$

56 Exploratory Analysis > Two-variable data analysis · Level 3

For a class project, a student surveyed 125 cars in a high school parking lot. The student recorded whether each car had a student or a staff tag and whether each was an American or a foreign model car. Of the 50 American cars, 20 had staff tags. Given that whether the car has a student or a staff tag and whether it is an American or a foreign model are independent, how many of the foreign cars had student tags?

A

20

B

30

C

45

D

50

E

75

57 Exploratory Analysis > Two-variable data analysis · Level 3

Men and women were sampled as to whether, in general, they feel "always rushed," "sometimes rushed," or "almost never rushed." A summary of those responses are given in the following table.

	Men	Women	Total
Always	44	95	139
Sometimes	79	69	148
Almost never	22	28	50
Total	145	192	337

Based on this table, which of the following statements is incorrect?

A

Most men feel "sometimes rushed."

B

Most women feel "always rushed."

C

Men are more likely than women to say that they are "almost never rushed."

D

More men than women say they are "sometimes rushed."

E

The proportion of people who are either "sometimes rushed" or "almost never rushed" is $1 - \dfrac{139}{337}$.

58 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

A simple random sample (SRS) of beachfront condo listings in Myrtle Beach comparing weekly rent ($) versus size (ft²) yields the following computer output. (a) Is a linear model appropriate for these data? Explain. (b) Interpret the slope of the regression line. (c) Interpret $r^2$ in context.

59 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

The calories and fat content per serving size of 10 brands of potato chips are fitted with a least squares regression line with computer output. (a) Is a line an appropriate model? Explain. (b) Interpret the slope of the regression line in context. (c) Interpret the y-intercept of the regression line in context. (d) What are the predicted calories for a brand with 10 g of fat per serving? (e) What are the actual calories for the brand with 10 g of fat per serving?

60 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

A research group believes it can predict a subject's ESP (telepathy and clairvoyance) testing result based on the average SAT math and verbal scores of those given ESP testing. The research group suspects that gender also makes a difference. A stratified random sample of high school students who have taken the SATs are given ESP testing. Below are two least squares analyses relating ESP test results to average SAT math and verbal scores for men and for women. A student has an average SAT math and verbal score of 689 and then receives an ESP test result of 84. What would you guess is the student's gender? Justify your answer.

61 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

The following is a scatterplot of the number of hurricanes hitting a Caribbean island each year after 1982 ($t$ gives years since 1982). (a) Draw a histogram of the frequencies of the number of hurricanes. (b) Name a feature apparent in the scatterplot but not in the histogram. (c) Name a feature shown by the histogram but not as obvious in the scatterplot.

62 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

Data are collected on the distance (in feet) that students sit from the front of the class and their exam averages for the class (on a 0–100 scale). A scatterplot of exam average $y$ versus distance $x$ from the front is described as linear, negative, and strong. (a) In the above context, what is meant by linear, by negative, and by strong? The regression equation is $\hat{y} = 97.5 - 1.15x$. (b) Interpret the slope in context. (c) One student who sat 12 feet from the front had a residual of -4.7. What was that student's exam average?

63 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

It is believed that customers will hesitate buying a luxury item if it seems to be priced too high or too low. A new-model boat is offered at different prices in eight different showrooms. The number sold versus price (\$1,000) is fitted with a least squares regression line, yielding the following summary statistics and residual plot. (a) Comment on the use of a linear model for this data. (b) How can this computer output be used to recommend the best cost to achieve the most sales? Explain.

64 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

The table below gives weight loss (in pounds) during the first month for six overweight patients on varying dosages of an experimental drug.

Dosage (grams), $x$	0.5	1.0	1.5	2.0	2.5	3.0
Weight Loss (pounds), $y$	7	10	12	14	16	17

Linear regression lines on $(x, y)$, on the transformed data $(x, \log y)$, and on the transformed data $(\sqrt{x}, y)$ result in the following computer output, respectively. (a) Interpret the coefficient of determination for the transformed data $(\sqrt{x}, y)$. (b) Compare the three regression lines as to goodness-of-fit for a linear model.

65 Exploratory Analysis > Two-variable data analysis (FR) · Level 3

In a study of salaries of young engineers at a software company, two regression analyses are run, the first on salary versus years of college education and the second on salary versus age. The graphs of the residuals follow. (a) Which of the regression lines indicates a better linear fit? Explain. (b) The two regression lines are used to find estimates for the salary of a 40-year-old software engineer with 4 years of college education. One of the engineers in the sample is 40 years old and has 4 years of college education. Which of the estimates are underestimates and which are overestimates of the salary of this engineer? Explain.

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