Question 1

The degrees of freedom for the test statistic for the slope is ____________________.

Accepted Answer

The answer of The degrees of freedom for the test...

Question 2

If the coefficient of correlation is 0.90,then the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is:
A) 90%
B) 81%
C) 95%
D) None of these choices.

Accepted Answer

The coefficient of correlation, r, squared gives the coefficient of determination, r^2, which represents the percentage of variation in the dependent variable (y) that is explained by the variation in the independent variable (x). Therefore, r^2 = 0.90^2 = 0.81, which means that 81% of the variation in y is explained by the variation in x. So, the correct choice is B.

Question 3

In a simple linear regression problem,r and b₀: A) must be equal to each other. B) must have the same sign. C) must have opposite signs. D) are not related.

Accepted Answer

In simple linear regression, $r$ (the correlation coefficient) and $b$ (the slope of the regression line) must have the same sign. This means if the relationship between the variables is positive, both $r$ and $b$ will be positive, indicating a positive relationship. Conversely, if the relationship is negative, both $r$ and $b$ will be negative, indicating a negative relationship.

Question 4

Truck Speed and Gas Mileage
An economist wanted to analyze the relationship between the speed of a truck (x)and its gas mileage (y).As an experiment a truck is operated at several different speeds and for each speed the gas mileage is measured.These data are shown below.
 $$\begin{array} { | l | l l l l l l l | } 
\hline \text { Speed } & 25 & 35 & 45 & 50 & 60 & 65 & 70 \
\hline \text { Gas Mileage } & 40 & 39 & 37 & 33 & 30 & 27 & 25 \
\hline
\end{array}$$
-{Truck Speed and Gas Mileage Narrative} Calculate the standard error of estimate,and describe what this statistic tells you about the regression line.

Accepted Answer

The answer of Truck Speed and Gas Mileage
An economist wanted...

Question 5

If the standard error of estimate is zero,then:
A) the coefficient of determination must be 1.0.
B) all the points fall on the regression line.
C) there is no unexplained variation left.
D) All of these choices are true.

Accepted Answer

When the standard error of estimate is zero, it means there is perfect prediction with no errors between the observed and predicted values. This implies that all points fall exactly on the regression line, leaving no unexplained variation. Consequently, the coefficient of determination, which measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s), would be exactly 1.0, indicating perfect predictability. Therefore, all of these statements are true under this condition.

Question 6

In simple linear regression,the coefficient of correlation r and the least squares estimate b₁ of the population slope $\beta$₁: A) must be equal. B) must have the same sign. C) are not related. D) None of these choices.

Accepted Answer

The coefficient of correlation r and the least squares estimate b1 of the population slope must have the same sign because both indicate the direction of the relationship between the independent and dependent variables.

Question 7

For a regression analysis to be valid,the error variable must have a standard deviation that is ____________________ regardless of the value of x.

Accepted Answer

The answer of For a regression analysis to be valid,the...

Question 8

The unbiased estimator of the variance of the error variable is found by taking ____________________ divided by n - 2.

Accepted Answer

The answer of The unbiased estimator of the variance of...

Question 9

For a regression analysis to be valid,the error variable must have a mean of ____________________.

Accepted Answer

The answer of For a regression analysis to be valid,the...

Question 10

In performing a regression analysis which of the following must be true about the distribution of the error variable?
A) The distribution is normal with mean zero.
B) The errors associated with one y value are independent of errors associated with another y value.
C) The standard deviation is constant for each value of x.
D) All of these choices are true.

Accepted Answer

All of the choices must be true for a valid regression analysis.

A) The error variable should have a normal distribution with a mean of zero to ensure that the residuals (observed values minus predicted values) are normally distributed, which is a requirement for many statistical tests and assumptions.

B) The errors associated with one y value should be independent of errors associated with another y value. This means that there should be no pattern or correlation in the residuals. If there is a pattern or relationship between the residuals, it suggests that there may be another variable that is influencing the outcome.

C) The standard deviation of the error variable should be constant for each value of x. This assumption is known as homoscedasticity, which means that the variance of the residuals should be the same (constant) for all levels of the predictor variable. If the variance of the residuals changes with respect to the predictor variable, this is known as heteroscedasticity, which can violate the assumptions of regression analysis.

Question 11

The coefficient of ____________________ measures the amount of variation in the dependent variable that is explained by the variation in the independent variable.

Accepted Answer

The answer of The coefficient of ____________________ measures the amount...

Question 12

If the regression line is horizontal,the slope is ____________________ and x and y are not related.

Accepted Answer

The answer of If the regression line is horizontal,the slope...

Question 13

For a regression analysis to be valid,the error variable must have a(n)____________________ distribution.

Accepted Answer

The answer of For a regression analysis to be valid,the...

Question 14

If the standard error of estimate is ____________________,this implies that the model's fit is poor.

Accepted Answer

The answer of If the standard error of estimate is...

Question 15

If the sum of squared residuals is zero,then the:
A) coefficient of determination must be 1.0.
B) coefficient of correlation must be 1.0.
C) linear relationship between x and y is perfect.
D) All of these choices are true.

Accepted Answer

If the sum of squared residuals is zero, it means that there is no deviation between the predicted and actual values, i.e., the regression line perfectly fits the data points. Thus, the coefficient of determination (which measures the proportion of the total variation in y that is explained by the regression model) must be 1.0, and the coefficient of correlation (which measures the strength and direction of the linear relationship between x and y) must also be 1.0. Therefore, all of the choices are true.

Question 16

If the regression line is horizontal,then we conclude that y ____________________ (is/is not)related to x.

Accepted Answer

The answer of If the regression line is horizontal,then we...

Question 17

For a regression analysis to be valid,the value of the error variable associated with any particular value of y is ____________________ of the value of the error variable associated with any other value of y.

Accepted Answer

The answer of For a regression analysis to be valid,the...

Question 18

The standard error of the estimate is a measure of the:
A) total variation in the y variable.
B) variation around the regression line.
C) percentage of variation in y explained by the variation in x.
D) the variation of the x variable.

Accepted Answer

The standard error of the estimate is a measure of the variation around the regression line, indicating how much the predicted values differ from the actual values.

Question 19

In simple linear regression,which of the following statements indicates there is no linear relationship between the variables x and y?
A) Coefficient of determination is -1.0.
B) Coefficient of correlation is 0.0.
C) Sum of squares for error is 0.0.
D) None of these choices.

Accepted Answer

A coefficient of correlation of 0.0 indicates no linear relationship between the variables. A coefficient of determination of 1.0 indicates a perfect linear relationship, and a sum of squares for error of 0.0 would indicate no error in the regression, which is highly unlikely.

Question 20

Which of the following assumptions concerning the probability distribution of the random error term is stated incorrectly?
A) The distribution is normal.
B) The mean of the distribution is 0.
C) The variance of the distribution increases as x increases.
D) The errors are independent from one value of y to the next.

Accepted Answer

The assumption that the variance of the distribution increases as x increases is incorrect. The error term is assumed to have a constant variance, which is known as homoscedasticity.

Quiz 16: A: Simple Linear Regression and Correlation

The degrees of freedom for the test statistic for the slope is ____________________.

If the coefficient of correlation is 0.90,then the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is:

In a simple linear regression problem,r and b₀:

If the standard error of estimate is zero,then:

In simple linear regression,the coefficient of correlation r and the least squares estimate b₁ of the population slope $\beta$ ₁:

For a regression analysis to be valid,the error variable must have a standard deviation that is ____________________ regardless of the value of x.

The unbiased estimator of the variance of the error variable is found by taking ____________________ divided by n - 2.

For a regression analysis to be valid,the error variable must have a mean of ____________________.

In performing a regression analysis which of the following must be true about the distribution of the error variable?

The coefficient of ____________________ measures the amount of variation in the dependent variable that is explained by the variation in the independent variable.

If the regression line is horizontal,the slope is ____________________ and x and y are not related.

For a regression analysis to be valid,the error variable must have a(n)____________________ distribution.

If the standard error of estimate is ____________________,this implies that the model's fit is poor.

If the sum of squared residuals is zero,then the:

If the regression line is horizontal,then we conclude that y ____________________ (is/is not)related to x.

For a regression analysis to be valid,the value of the error variable associated with any particular value of y is ____________________ of the value of the error variable associated with any other value of y.

The standard error of the estimate is a measure of the:

In simple linear regression,which of the following statements indicates there is no linear relationship between the variables x and y?

Which of the following assumptions concerning the probability distribution of the random error term is stated incorrectly?

Quiz 16: A: Simple Linear Regression and Correlation

The degrees of freedom for the test statistic for the slope is ____________________.

If the coefficient of correlation is 0.90,then the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is:

In a simple linear regression problem,r and b0:

If the standard error of estimate is zero,then:

In simple linear regression,the coefficient of correlation r and the least squares estimate b1 of the population slope β\betaβ 1:

For a regression analysis to be valid,the error variable must have a standard deviation that is ____________________ regardless of the value of x.

The unbiased estimator of the variance of the error variable is found by taking ____________________ divided by n - 2.

For a regression analysis to be valid,the error variable must have a mean of ____________________.

In performing a regression analysis which of the following must be true about the distribution of the error variable?

The coefficient of ____________________ measures the amount of variation in the dependent variable that is explained by the variation in the independent variable.

If the regression line is horizontal,the slope is ____________________ and x and y are not related.

For a regression analysis to be valid,the error variable must have a(n)____________________ distribution.

If the standard error of estimate is ____________________,this implies that the model's fit is poor.

If the sum of squared residuals is zero,then the:

If the regression line is horizontal,then we conclude that y ____________________ (is/is not)related to x.

For a regression analysis to be valid,the value of the error variable associated with any particular value of y is ____________________ of the value of the error variable associated with any other value of y.

The standard error of the estimate is a measure of the:

In simple linear regression,which of the following statements indicates there is no linear relationship between the variables x and y?

Which of the following assumptions concerning the probability distribution of the random error term is stated incorrectly?

In a simple linear regression problem,r and b₀:

In simple linear regression,the coefficient of correlation r and the least squares estimate b₁ of the population slope $\beta$ ₁: