Question 1

The proportion of the variation in the dependent variable y that is explained by the estimated regression equation is measured by the​
A) ​correlation coefficient.
B) ​standard error of the estimate.
C) ​coefficient of determination.
D) ​confidence interval estimate.

Accepted Answer

The coefficient of determination, also known as R-squared, measures how much of the variation in the dependent variable is explained by the independent variable(s) in the regression equation. It ranges between 0 and 1, with higher values indicating a stronger relationship between the variables. A value of 1 indicates that the regression line perfectly fits the data, while a value of 0 indicates that the regression line does not explain any of the variation in the data.

Question 2

In regression analysis, which of the following assumptions is not true about the error term ε?
A) The expected value of the error term is one.
B) The variance of the error term is the same for all values of x.
C) The values of the error term are independent.
D) The error term is normally distributed.

Accepted Answer

The expected value of the error term is assumed to be zero, not one. The other three assumptions (equal variance, independence, and normality of the error term) are true assumptions of regression analysis.

Question 3

The coefficient of correlation
A) is the square of the r-square.
B) is the square root of the r-square.
C) can never be equal to r-square.
D) can never be negative.

Accepted Answer

The coefficient of correlation is equal to the square root of the r-square.

Question 4

In simple linear regression, r² is the A) mean square error. B) correlation coefficient. C) squared residual. D) coefficient of determination.

Accepted Answer

The answer of In simple linear regression, r² is the A)...

Question 5

The mathematical equation relating the independent variable to the expected value of the dependent variable; that is, E(y) = β₀ + β₁x, is known as the A) regression equation. B) correlation model. C) estimated regression equation. D) regression model.

Accepted Answer

The answer of The mathematical equation relating the independent variable...

Question 6

The coefficient of determination
A) cannot be negative.
B) is the square root of the coefficient of correlation.
C) is the same as the coefficient of correlation.
D) can be negative or positive.

Accepted Answer

The coefficient of determination, often denoted as R^2, represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It is a measure of the goodness of fit of a model and ranges from 0 to 1, meaning it cannot be negative.

Question 7

The standard error of the estimate is the
A) standard deviation of t.
B) square root of SSE.
C) square root of SST.
D) square root of MSE.

Accepted Answer

The standard error of the estimate is the square root of the mean squared error (MSE), which is calculated by dividing the sum of squares error (SSE) by the degrees of freedom. Therefore, the correct choice is D, the square root of MSE.

Question 8

In regression analysis, the unbiased estimate of the variance is
A) coefficient of correlation.
B) coefficient of determination.
C) mean square error.
D) slope of the regression equation.

Accepted Answer

The unbiased estimate of the variance is found by calculating the mean square error, which measures the average distance between the predicted values and the actual values in the regression analysis. The coefficient of correlation and determination measure the strength of the relationship between the variables, while the slope of the regression equation represents the change in the dependent variable for every one-unit change in the independent variable. None of these directly provides an estimate of the variance.

Question 9

In regression analysis, the model in the form y = $\beta _ { 0 }$ + $\beta _ { 1 }$ x + $\varepsilon$ is called the
A) regression equation.
B) correlation model.
C) estimated regression equation.
D) regression model

Accepted Answer

The answer of In regression analysis, the model in the...

Question 10

The model developed from sample data that has the form of $\hat { y }$ = b₀ + b₁x is known as the A) regression equation. B) correlation model. C) estimated regression equation. D) regression model.

Accepted Answer

The answer of The model developed from sample data that...

Question 11

If only MSE is known, you can compute the
A) r-square.
B) coefficient of correlation.
C) standard error.
D) ith residual.

Accepted Answer

The standard error can be computed using MSE and is a measure of the variability of the sample mean from the true population mean. The coefficient of correlation and r-square cannot be computed solely from MSE, as they require information about the relationship between two variables. The ith residual also cannot be determined from only MSE, as it requires information about the individual data points.

Question 12

The interval estimate of the mean value of y for a given value of x is the
A) prediction interval estimate.
B) confidence interval estimate.
C) average regression interval.
D) x versus y correlation interval.

Accepted Answer

The interval estimate of the mean value of y for a given value of x is a confidence interval estimate. This means we are estimating a range of values where the true population mean of y for a given value of x is likely to fall, with a certain level of confidence. The other options are not correct definitions or terms for this type of interval estimate.

Question 13

The value of the coefficient of correlation (r) A) can be equal to the value of the coefficient of determination (r²). B) can never be equal to the value of the coefficient of determination (r²). C) is always smaller than the value of the coefficient of determination (r²). D) is always larger than the value of the coefficient of determination (r²).

Accepted Answer

The coefficient of correlation (r) is the square root of the coefficient of determination (r squared or r^2), so it can be equal to the value of the coefficient of determination (r^2). Therefore, option A is correct. Option B is incorrect because r can be equal to r^2. Option C and D are incorrect because r is always smaller than or equal to r^2, not always larger or smaller.

Question 14

A regression analysis between sales (y in $1000) and advertising (x in dollars) resulted in the following equation:   $\hat { y }$ = 30,000 + 4x
 
The above equation implies that an
A) increase of $4 in advertising is associated with an increase of $4000 in sales.
B) increase of $1 in advertising is associated with an increase of $4 in sales.
C) increase of $1 in advertising is associated with an increase of $34,000 in sales.
D) increase of $1 in advertising is associated with an increase of $4000 in sales.

Accepted Answer

The answer of A regression analysis between sales (y in...

Question 15

In the following estimated regression equation $\hat { y }$ = b₀ + b₁x, A) b₁ is the slope. B) b₁ is the intercept. C) b₁x is the slope. D) b₁x is the intercept.

Accepted Answer

b₁ is the slope because it represents the change in the dependent variable (y) for a one-unit change in the independent variable (x).

Question 16

Regression analysis is a statistical procedure for developing a mathematical equation that describes how
A) one independent and one or more dependent variables are related.
B) several independent and several dependent variables are related.
C) one dependent and one or more independent variables are related.
D) one dependent, one independent, and several error variables are related.

Accepted Answer

Regression analysis is used to determine how one dependent variable is related to one or more independent variables. The goal is to develop a mathematical equation that can predict the value of the dependent variable based on the values of the independent variables.

Question 17

If the coefficient of determination is a positive value, then the coefficient of correlation
A) must also be positive.
B) must be zero.
C) can be either positive or negative.
D) can be larger than 1.

Accepted Answer

The coefficient of determination (R^2) indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s), but it does not indicate the direction of the relationship. The coefficient of correlation (r) indicates both the strength and the direction of the linear relationship between two variables. Since R^2 is the square of r, if R^2 is positive, r can be either positive or negative, reflecting either a direct or inverse relationship, respectively.

Question 18

The interval estimate of an individual value of y for a given value of x is the
A) prediction interval estimate.
B) confidence interval estimate.
C) average regression interval.
D) x versus y correlation interval.

Accepted Answer

The prediction interval estimate is used to provide a range of values where an individual value of y is likely to fall, given a specific value of x. The confidence interval estimate, on the other hand, provides a range of values where the population mean of y is likely to fall for a specific value of x. The other options, "average regression interval" and "x versus y correlation interval," are not commonly used statistical terms.

Question 19

In regression analysis, the error term $\varepsilon$is a random variable with a mean or expected value of
A) 0.
B) 1.
C) $\mu$.
D) $\bar { x }$ .

Accepted Answer

In regression analysis, the error term $\varepsilon$ is assumed to have an expected value of 0, indicating that the errors are randomly distributed around the regression line.

Question 20

In a regression analysis, the standard error of the estimate is determined to be 4.In this situation, the MSE
A) is 2.
B) is 16.
C) depends on the sample size.
D) requires a known SSE.

Accepted Answer

The mean squared error (MSE) is equal to the standard error of estimate (SE) squared. Therefore, MSE = 4^2 = 16.

Quiz 14: Simple Linear Regression

The proportion of the variation in the dependent variable y that is explained by the estimated regression equation is measured by the

In regression analysis, which of the following assumptions is not true about the error term ε?

The coefficient of correlation

In simple linear regression, r² is the

The mathematical equation relating the independent variable to the expected value of the dependent variable; that is, E(y) = β₀ + β₁x, is known as the

The coefficient of determination

The standard error of the estimate is the

In regression analysis, the unbiased estimate of the variance is

In regression analysis, the model in the form y = $\beta _ { 0 }$ + $\beta _ { 1 }$ x + $\varepsilon$ is called the

The model developed from sample data that has the form of $\hat { y }$ = b₀ + b₁x is known as the

If only MSE is known, you can compute the

The interval estimate of the mean value of y for a given value of x is the

The value of the coefficient of correlation (r)

A regression analysis between sales (y in $1000) and advertising (x in dollars) resulted in the following equation: $\hat { y }$ = 30,000 + 4x The above equation implies that an

In the following estimated regression equation $\hat { y }$ = b₀ + b₁x,

Regression analysis is a statistical procedure for developing a mathematical equation that describes how

If the coefficient of determination is a positive value, then the coefficient of correlation

The interval estimate of an individual value of y for a given value of x is the

In regression analysis, the error term $\varepsilon$ is a random variable with a mean or expected value of

In a regression analysis, the standard error of the estimate is determined to be 4.In this situation, the MSE