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?

Accepted Answer

To find the predicted difference in GPA, we use the regression equation Y' = 0.0016X + 0.6. For Karen, with an SAT score of 1800, her predicted GPA is Y' = 0.0016(1800) + 0.6 = 2.88. For Mary, with an SAT score of 1600, her predicted GPA is Y' = 0.0016(1600) + 0.6 = 2.56. The difference is 2.88 - 2.56 = 0.32, so Karen's predicted GPA is 0.32 higher than Mary's predicted GPA.

Question 2

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

Accepted Answer

The slope in a simple linear regression equation represents the change in the dependent variable (Y) for a one-unit change in the independent variable (X). Therefore, if the slope is -0.002, it means that Y will decrease by 0.002 units for every one-unit increase in X.

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?

Accepted Answer

The coefficient of X (height) in the regression equation Y' = 4X - 136 indicates the amount of change in the dependent variable (weight) for a one-unit change in the independent variable (height). Therefore, for each additional inch in height, the weight is expected to increase by 4 lbs.

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?

Accepted Answer

Sarah was likely extrapolating beyond the range of the data used to create the regression model, leading to an unrealistic prediction.

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?

Accepted Answer

Since the variance of Y is 0, it means Y is a constant. Therefore, the variance of the predicted values of Y is also 0.

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?

Accepted Answer

The correlation coefficient $\rho_{XY} = 0$ indicates no linear relationship between X and Y, leading to a slope of zero in the regression model.

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?

Accepted Answer

The slope of the regression line is given by:$$b = r_{XY} \frac{s_Y}{s_X}$$Substituting the given values, we get:$$b = 0.5 \frac{1}{1} = 0.5$$Therefore, the slope of the regression line is 0.5 or -0.5.

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?

Accepted Answer

The residual score is the difference between the actual score and the predicted score. The predicted score for Jamie, who watches TV two hours per day, is Y' = -7(2) + 100 = -14 + 100 = 86. Jamie's actual score is 91. Therefore, the residual score is 91 - 86 = 5.

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.

Accepted Answer

Model 2 is preferred because it has a higher R-squared value (0.65) compared to model 1 (0.014), indicating a stronger relationship between the variables and thus a better fit for predicting Y.

Question 10

The standardized regression slope ( $b_{Y X}^{*}$ )

Accepted Answer

The standardized regression slope ( $b_{Y X}^{*}$ ) may never be greater than +1.00 because it is a measure of the strength of the linear relationship between two variables, and the strength of a linear relationship can never be greater than 1.00.

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?

Accepted Answer

Residual scores are the differences between observed and predicted values of Y. If two individuals have the same observed score on Y, their residuals will be equal only if they have the same score on X (since the prediction is based on X) or if the slope of the relationship between X and Y is zero (meaning changes in X do not affect Y, so the prediction for Y is the same for all values of X).

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?

Accepted Answer

The proportion of variation in Y that is not predictable from X is given by 1 - r^2. Since r = .3, r^2 = .09, and 1 - r^2 = 1 - .09 = .91.

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?

Accepted Answer

The answer of Bob and Brian both used simple linear...

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?

Accepted Answer

The answer of In the scenario described in Question 13,...

Question 15

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

Accepted Answer

The assumption is that the errors have a mean of 0, but not necessarily a variance of 1. The variance should be constant but not specifically 1.

Question 16

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

Accepted Answer

The answer of In which of the following situations is...

Question 17

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

Accepted Answer

When the slope of the estimated regression line is positive, it indicates that as X increases, Y also increases, which directly implies a positive correlation between X and Y.

Question 18

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

Accepted Answer

When r_XY = 1, it indicates a perfect positive linear relationship, meaning all points will fall exactly on the regression line.

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_X_Y = - 0.2, $\bar{X}$ = 40, $\bar{Y}$ = 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

Accepted Answer

The correlation coefficient (r) between the age of the house (X) and the selling price (Y) is -0.2, indicating a negative relationship. This means as the age of the house increases, the selling price tends to decrease. Since Derek's house, at 50 years, is older than the average age of houses in the study ($\bar{X}$ = 40 years), the predicted selling price of his house would be less than the average selling price ($\bar{Y}$ = 460,000 dollars).

Question 20

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

Accepted Answer

The unstandardized regression line in simple linear regression always passes through the point ($\bar{X}$, $\bar{Y}$), where $\bar{X}$ is the mean of the independent variable and $\bar{Y}$ is the mean of the dependent variable. This is a property of the regression line that ensures it best fits the data according to the least squares criterion.

Quiz 17: Simple Linear Regression

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?

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

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?

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?

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?

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?

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?

The standardized regression slope ( $b_{Y X}^{*}$ )

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

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?

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?

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

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

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

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

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

Quiz 17: Simple Linear Regression

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?

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

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?

It is known that μ\muμ X = 1.5, σ\sigmaσ X2 = 25, μ\muμ Y = 10, σ\sigmaσ Y2 = 0. A simple linear regression model was estimated. Which of the following is the variance of the predicted values of Y?

It is known that μ\muμ X = 10, σ\sigmaσ X2 = 16, μ\muμ Y = 52, σ\sigmaσ Y2 = 8, ρ\rhoρ XY = 0. A simple linear regression model is estimated. Which of the following statements is true?

It is known that rXY = 0.5, sX2 = 1, sY2 = 1. A simple linear regression model is estimated. The regression line will have a slope of which one of the following?

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?

The standardized regression slope ( bYX∗b_{Y X}^{*}bYX∗​ )

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

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

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?

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

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

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

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

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

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?

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?

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?

The standardized regression slope ( $b_{Y X}^{*}$ )

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?

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