Deck 17: Simple Linear Regression

Question 1

The regression line for predicting college GPA from SAT scores is found to be Y' = 0.0016X + 0.6. Karen's SAT score is 1800, and Mary's SAT score is 1600. What is the predicted difference in their college GPA?

A) Karen's predicted GPA is 0.32 higher than Mary's predicted GPA.
B) Karen's predicted GPA is 0.92 higher than Mary's predicted GPA.
C) Karen's predicted GPA is 0.32 lower than Mary's predicted GPA.
D) Karen's predicted GPA is 0.32 lower than Mary's predicted GPA.
E) Karen and Mary have the same predicted GPA.

Answer 1

A

Question 2

In simple linear regression, if the slope is found to be -0.002,

A) the value of Y is equal to -.002 when X is 0.
B) the value of Y is equal to -.002 when X is 1.
C) the value of Y will decrease by 0.002 units when X increases by 1 unit.
D) there is a negative, but very weak relationship between X and Y.

Answer 2

C

Question 3

Sarah collected the data on heights and weights from 100 graduate students. Based on the data, she built a simple linear regression model to predict weight (in lbs) from height (in inches). The regression line is found to be Y' = 4X - 136. Which of the following statements is the correct interpretation of the equation?

A) When weight increases by 1 lb, height is expected to increase by 4 inches.
B) When weight decreases by 4 lbs, height is expected to increase by 1 inch.
C) When height increases by 1 inch, weight is expected to increase by 4 lbs.
D) When height increases by 4 inches, weight is expected to decrease by 1 lb.

Answer 3

C

Question 4

In the scenario as described in Question 3, Sarah now used the regression line she obtained to predict the weight of her three-year old niece, who is 34 inches tall. The predicted weight for her niece, however, turned out to be 0 lb. What is the problem with Sarah's prediction?

A) Sarah must have incorrectly estimated the intercept of the regression line.
B) The regression line was correct, but a computational error was made in the prediction.
C) Sarah should have included the error term (e_i) in the regression model when she made the prediction.
D) Sarah was extrapolating beyond the sample predictor data.
E) There is no problem with Sarah's prediction. This is how statistics work.

Question 5

It is known that $\mu$ _X = 1.5, $\sigma$ _X² = 25, $\mu$ _Y = 10, $\sigma$ _Y² = 0. A simple linear regression model was estimated. Which of the following is the variance of the predicted values of Y?

A) 0
B) 15
C) 25
D) 250
E) It depends on the correlation between X and Y.

Question 6

It is known that $\mu$ _X = 10, $\sigma$ _X² = 16, $\mu$ _Y = 52, $\sigma$ _Y² = 8, $\rho$ _XY = 0. A simple linear regression model is estimated. Which of the following statements is true?

A) X and Y are completely unrelated.
B) The slope of the regression model will be zero.
C) The intercept of the regression model will be zero.
D) The prediction of Y based on the linear model will be very accurate.

Question 7

It is known that r_XY = 0.5, s_X² = 1, s_Y² = 1. A simple linear regression model is estimated. The regression line will have a slope of which one of the following?

A) 0
B) 0.5
C) 0.5 or -0.5
D) 1 or -1
E) It depends on the values of and .

Question 8

In a study of the relation between hours watching TV per day (X) and scores on the final exam (Y), the equation of regression line is found to be Y' = -7X + 100. Suppose Jamie watches TV two hours per day, and he scored a 91 on the exam. What is the residual score for Jamie?

A) -7
B) -5
C) 0
D) 5
E) 100

Question 9

Doug wanted to use simple linear regression to study the relation between the time to complete a marathon (in hours) (Y) and the fluid intake (in ml) during the race (X). Based on the same data set, he estimated two models.
Model 1: X₁ = total amount of fluid intake; Y = .00028X₁ + 3.97. R₁² = .014.
Model 2: X₂ = amount of fluid intake per hour; Y = -.0052X₂ + 7.84. R₂² = .65.
Suppose for both models, all assumptions for linear regression are satisfied. Compare the two models.

A) Doug should use model 1, because the correlation between X₁ and Y is stronger than that between X₂ and Y (b_YX1 > b_YX2).
B) Doug should use model 2, because it gives more accurate prediction of Y (the finishing time) than model 1 (R₁² < R₂²).
C) Doug should use model 1, because model 2 will give unreasonable (negative) predicted values of Y when X is large.
D) Both models are problematic, because the units of X are different from the unit of Y.

Question 10

The standardized regression slope ( )

A) may never be negative.
B) may never be greater than +1.00.
C) is always equal to 0.
D) None of the above.

Question 11

If two individuals have the same observed score on the dependent variable Y, their residual scores will be which one of the following?

A) Always be equal.
B) Be equal only when they have the same score on X.
C) Be equal only when the slope equals zero.
D) Both b and c

Question 12

In simple linear regression, if r_XY = .3, the proportion of variation in Y that is not predictable from X is which one of the following?

A) 0.09
B) 0.3
C) 0.7
D) 0.91
E) It depends on the slope.

Question 13

Bob and Brian both used simple linear regression to predict the consumption of ice cream (ml/person) (Y) based on temperature (°F) (X). However, they used two different data sets to estimate the model: Bob's sample includes only children younger than 12 (r_XY = 0.6), while Brian's sample includes only adult consumers (r_XY = 0.4). Which of the following statements is always true?

A) Bob and Brian will get different estimates of intercept.
B) Bob and Brian will get different estimates of slope.
C) Bob and Brian will get different prediction equations.
D) Bob and Brian will get different R² for their models.

Question 14

In the scenario described in Question 13, suppose Bob and Brian have both converted their data to z score scale and estimated regression models using the standardized scores. Which of the following statements is false?

A) Bob and Brian will get different estimates of intercept.
B) Bob and Brian will get different estimates of slope.
C) Bob and Brian will get different prediction equations.
D) Bob and Brian will get different R² for their model.

Question 15

The assumptions of the simple linear regression model do not include:

A) The errors are normally distributed.
B) The errors have constant variance across different values of X.
C) The errors are independent of each other.
D) The errors have mean of 0 and variance of 1.

Question 16

In which of the following situations is it most appropriate to use the simple linear regression model?

A)
B)
C)
D)

Question 17

If the slope of the estimated regression line is positive, the correlation between X and Y

A) must be positive.
B) must be negative.
C) may be zero.
D) depends on the mean and variance of X and Y.

Question 18

If r_XY = 1, which of the following statements is true?

A) All points will fall on the regression line.
B) The regression line will have an intercept of 0.
C) The regression line will have a slope of 1.
D) The prediction of future observations will be 100% accurate.

Question 19

Derek is studying the relation between the selling price of a house (in dollars) (Y) and the age of the house (in years) (X). It is shown that r_XY = - 0.2, = 40, = 460,000. If Derek's own house was constructed 50 years ago, then the predicted selling price of his house based on simple linear regression would be

A) more than 460,000 dollars.
B) less than 460,000 dollars.
C) exactly 460,000 dollars.
D) impossible to be determined based on the information given.

Question 20

In simple linear regression, the unstandardized regression line will always pass

A) at least one data point.
B) at least two data points.
C) the point ( , ).
D) the point (0,0).

Question 21

Which assumption(s) involved in simple linear regression can be assessed by examining the residual plot (e_i vs. X_i)?

A) Independence
B) Homogeneity
C) Linearity
D) All of the above

Question 22

If the homogeneity assumption is violated, the possible consequences include

A) biased estimates of regression coefficients.
B) deflated standard error of estimates.
C) larger number of Type I errors.
D) nonnormal conditional distribution of Y.

Question 23

In simple linear regression, the assumption of normality states that

A) the observed scores on Y are normally distributed.
B) the conditional distributions of Y are normal in shape.
C) the observed scores on X are normally distributed.
D) the distributions of regression coefficients are normal in shape.
E) there are no outliers.

Question 24

Dr. Guinea was studying the relation between the amount of caffeine intake and people's performance on a difficult task. He found out that as the amount of caffeine intake increases, the time to finish the task first decreases, and then increases. If he used the data to fit a linear regression model, which assumption would likely be violated?

A) Independence
B) Homogeneity
C) Linearity
D) Normality
E) Fixed X

Question 25

In a simple linear regression, if SS_res = 150 and SS_total = 200, what is the proportion of variation in Y that is predictable from X?

A) 25%
B) 42.86%
C) 56.25%
D) 75%

Question 26

You are given the following pairs of scores on X (Pretest score) and Y (Posttest score).
$\begin{array}{cc}\hline X & Y \\\hline 65 & 74 \\82 & 87 \\70 & 82 \\46 & 53 \\55 & 69 \\75 & 81 \\\hline\end{array}$
a. Find the linear regression model for predicting Y from X.
b. Use the prediction model obtained to predict the value of Y for a new person who scored 80 on the pretest.

Question 27

You are given the following pairs of scores on X (Percentage of students whose families are below poverty line) and Y (Percentage of students at or above proficiency level) for nine schools.
a. Find the linear regression model for predicting Y from X.
$\begin{array}{cc}\hline X & Y \\\hline 92.3 & 18.9 \\0.9 & 87.3 \\25.1 & 48.0 \\67.1 & 45.4 \\24.7 & 80.2 \\90.7 & 13.7 \\44.0 & 26.9 \\65.5 & 58.6 \\40.6 & 46.3 \\\hline\end{array}$
b. Use the prediction model obtained to predict the value of Y for a school that has 50% of students whose families are below the poverty line.

Question 28

The prediction equation for predicting Y (the amount of ice cream in pints consumed per person) from X (temperature in Fahrenheit) is Y' = 0.003X + 0.2. What is the observed mean for Y if $\mu$ _X = 70 and $\sigma$ _X² = 25?

Question 29

You are given the following pairs of scores on X (height in inches) and Y (weight in lbs).
Perform the following computations using $\alpha$ = .05.
$\begin{array}{cc}X & Y \\\hline 66 & 140 \\69 & 155 \\72 & 195 \\74 & 160 \\72 & 155 \\67 & 145 \\66 & 135 \\71 & 170 \\70 & 130 \\68 & 170 \\72 & 190 \\69 & 145 \\73 & 155 \\68 & 150 \\68 & 130 \\69 & 145 \\69 & 150 \\66 & 120 \\62 & 131 \\62 & 120 \\64 & 102 \\68 & 110 \\63 & 116 \\64 & 125 \\62 & 110 \\\hline\end{array}$
a. The regression equation of Y predicted by X.
b. Test of the significance of X as a predictor.
c. Plot Y versus X.
d. Compute the residuals.
e. Plot residuals versus X.

Question 30

Dr. Watt is studying the relation between the percentage of a population who has a bachelor's degree (X) and the average income (Y) in 108 cities. After fitting a simple linear regression model, he decides to assess whether the assumptions of the model are reasonably satisfied. Below is one of the plots he uses to assess the assumptions.
(a) What assumption(s) is Dr. Watt trying to assess using this plot?
(b) Based on the plot, is there any indication of assumption violations? If so, which assumption(s) has (have) been violated?
(c) What are the possible consequences of the assumption violation(s)?
(d) Suggest at least one solution to fix the problem.

Question 31

You are given the following pairs of scores on X (Pretest score) and Y (Posttest score).
a. Find the linear regression model for predicting Y from X.
$\begin{array}{cc}\hline X& Y \\\hline 65 & 74 \\82 & 87 \\70 & 82 \\46 & 53 \\55 & 69 \\75 & 81 \\\hline\end{array}$
b. Use the prediction model obtained to predict the value of Y for a new person who scored 80 on the pretest.

Question 32

You are given the following pairs of scores on X (Percentage of students whose families are below poverty line) and Y (Percentage of students at or above proficiency level) for nine schools.
a. Find the linear regression model for predicting Y from X.
$\begin{array}{cc}\hline X & Y \\\hline 92.3 & 18.9 \\0.9 & 87.3 \\25.1 & 48.0 \\67.1 & 45.4 \\24.7 & 80.2 \\90.7 & 13.7 \\44.0 & 26.9 \\65.5 & 58.6 \\40.6 & 46.3 \\\hline\end{array}$
b. Use the prediction model obtained to predict the value of Y for a school that has 50% of students whose families are below the poverty line.

Question 33

The prediction equation for predicting Y (the amount of ice cream in pints consumed per person) from X (temperature in Fahrenheit) is Y' = 0.003X + 0.2. What is the observed mean for Y if $\mu$ _X = 70 and $\sigma$ _X² = 25?

Question 34

You are given the following pairs of scores on X (height in inches) and Y (weight in lbs).
Perform the following computations using $\alpha$ = .05.
$\begin{array}{cc}X & Y \\\hline 66 & 140 \\69 & 155 \\72 & 195 \\74 & 160 \\72 & 155 \\67 & 145 \\66 & 135 \\71 & 170 \\70 & 130 \\68 & 170 \\72 & 190 \\69 & 145 \\73 & 155 \\68 & 150 \\68 & 130 \\69 & 145 \\69 & 150 \\66 & 120 \\62 & 131 \\62 & 120 \\64 & 102 \\68 & 110 \\63 & 116 \\64 & 125 \\62 & 110 \\\hline\end{array}$
a. The regression equation of Y predicted by X.
b. Test of the significance of X as a predictor.
c. Plot Y versus X.
d. Compute the residuals.
e. Plot residuals versus X.

Question 35

Dr. Watt is studying the relation between the percentage of a population who has a bachelor's degree (X) and the average income (Y) in 108 cities. After fitting a simple linear regression model, he decides to assess whether the assumptions of the model are reasonably satisfied. Below is one of the plots he uses to assess the assumptions.
(a) What assumption(s) is Dr. Watt trying to assess using this plot?
(b) Based on the plot, is there any indication of assumption violations? If so, which assumption(s) has (have) been violated?
(c) What are the possible consequences of the assumption violation(s)?
(d) Suggest at least one solution to fix the problem.