Question 1

If the error variable $\varepsilon$ is normally distributed, the test statistic for testing H₀: $\beta$₁ = 0 has a Student t-distribution with n -2 degrees of freedom.

Accepted Answer

This statement is true according to the properties of Student's t-distribution when testing the null hypothesis that the population regression coefficient is equal to zero. The test statistic follows a t-distribution with n-2 degrees of freedom, where n is the sample size.

Question 2

If the value of the sum of squares for error SSE equals zero, then the coefficient of determination must equal zero.

Accepted Answer

If the value of SSE equals zero, it means that the regression line perfectly fits all the data points, resulting in a coefficient of determination or R-squared value of 1, not 0.

Question 3

The value of the sum of squares for regression SSR can never be larger than the value of sum of squares for error SSE.

Accepted Answer

The value of the sum of squares for regression (SSR) can indeed be larger than the sum of squares for error (SSE) because SSR reflects the variation explained by the regression model, while SSE represents the unexplained variation. The total variation in the data (SST) is the sum of SSR and SSE, and there's no inherent constraint that SSR must be smaller than SSE.

Question 4

If the coefficient of determination is 0.95, this means that 95% of the y values were predicted correctly by the regression line.

Accepted Answer

The coefficient of determination (R-squared) is a measure of how well the regression line fits the data. It represents the proportion of the variance in the dependent variable (y) that can be explained by the independent variable(s) (x). A coefficient of determination of 0.95 means that 95% of the variation in the dependent variable can be explained by the independent variable(s), but it does not necessarily mean that 95% of the y values were predicted correctly by the regression line. The regression line may still have some error when predicting individual y values.

Question 5

A store manager gives a pre-employment examination to new employees. The test is scored from 1 to 100. He has data on their sales at the end of one year measured in dollars. He wants to know if there is any linear relationship between pre-employment examination score and sales. An appropriate test to use is the t-test of the population correlation coefficient.

Accepted Answer

The answer of A store manager gives a pre-employment examination...

Question 6

The value of the sum of squares for regression SSR can never be larger than the value of total sum of squares SST.

Accepted Answer

This is true because the total sum of squares (SST) represents the variation in the dependent variable around its mean, while the sum of squares for regression (SSR) represents the variation in the dependent variable that is explained by the regression model. Therefore, SSR can only account for a portion of the total variation in the dependent variable, and it is impossible for SSR to be larger than SST.

Question 7

In a simple linear regression model, testing whether the slope $\beta$₁ of the population regression line could be zero is the same as testing whether or not the population coefficient of correlation $\rho$ equals zero.

Accepted Answer

This is true since if the population coefficient of correlation equals zero, the slope of the population regression line will also be zero.

Question 8

A zero correlation coefficient between a pair of random variables means that there is no linear relationship between the random variables.

Accepted Answer

A zero correlation coefficient indicates that the two variables are not related in a linear fashion. However, it is still possible for there to be a non-linear relationship between the variables.

Question 9

If the coefficient of determination is 1.0, then the coefficient of correlation must be 1.0.

Accepted Answer

When the coefficient of determination is 1.0, it means that all the variation in the dependent variable is explained by the independent variable(s). In such a scenario, the coefficient of correlation between the two variables would be 1.0, indicating a perfect positive linear relationship between them. Therefore, the statement is true.

Question 10

The probability distribution of the error variable $\varepsilon$ is normal, with mean E($\varepsilon$) = 0, and standard deviation $\sigma$_{$\varepsilon$}=1.

Accepted Answer

The answer of The probability distribution of the error variable...

Question 11

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0.

Accepted Answer

The answer of If all the points in a scatter...

Question 12

If the coefficient of correlation is -0.81, then the percentage of the variation in y that is explained by the regression line is 81%.

Accepted Answer

The answer of If the coefficient of correlation is -0.81,...

Question 13

When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same, the standard error of estimate s_{$\varepsilon$} will be 0.0.

Accepted Answer

The standard error of estimate measures the average distance that the observed values fall from the regression line. If the actual and predicted values are the same, there is no deviation, resulting in a standard error of 0.

Question 14

A zero population correlation coefficient for x and y means that there is no type of relationship whatsoever between x and y.

Accepted Answer

A zero population correlation coefficient indicates there is no linear relationship between x and y, but they could still have a non-linear relationship.

Question 15

Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y.

Accepted Answer

Correlation analysis measures the strength and direction of a linear relationship between two variables. It determines the degree to which changes in one variable are associated with changes in the other variable.

Question 16

If there is no linear relationship between two variables x and y, the coefficient of determination must be -1.0.

Accepted Answer

The coefficient of determination, also known as R-squared, measures the proportion of the variation in the dependent variable y that is explained by the independent variable x. If there is no linear relationship between x and y, then the R-squared value will be 0, indicating that none of the variation in y is explained by x. Therefore, the statement in the question is false.

Question 17

If the coefficient of determination is 0.95, this means that 95% of the variation in the independent variable x can be explained by the y variable.

Accepted Answer

The coefficient of determination (R-squared) measures the proportion of variation in the dependent variable (y) that can be explained by the independent variable (x). Therefore, a coefficient of determination of 0.95 means that 95% of the variation in y can be explained by x, not the other way around.

Question 18

The coefficient of determination is equal to the coefficient of correlation squared.

Accepted Answer

This statement is true. The coefficient of determination (R^2) is equal to the square of the coefficient of correlation (R).

Question 19

In a simple linear regression problem, the least squares line is $\tilde { y } = - 3.75 + 1.25 x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be -0.90.

Accepted Answer

The coefficient of determination (R²) is 0.81, which means the absolute value of the correlation coefficient (r) is the square root of 0.81, i.e., 0.90. However, the sign of r is determined by the slope of the regression line, which is positive (1.25), so r should be +0.90, not -0.90.

Question 20

In simple linear regression, the denominator of the standard error of estimate s_{$\varepsilon$} is $\sqrt { n - 2 }$ .

Accepted Answer

The denominator of the standard error of estimate s_{$\varepsilon$} is $\sqrt { n - 2 }$ because it is a measure of the variability of the residuals, which are the differences between the observed values of the dependent variable and the values predicted by the regression line. The sample size n is reduced by 2 because the regression line is estimated using two parameters, the intercept and the slope.

Quiz 16: Simple Linear Regression and Correlation

If the error variable $\varepsilon$ is normally distributed, the test statistic for testing H₀: $\beta$ ₁ = 0 has a Student t-distribution with n -2 degrees of freedom.

If the value of the sum of squares for error SSE equals zero, then the coefficient of determination must equal zero.

The value of the sum of squares for regression SSR can never be larger than the value of sum of squares for error SSE.

If the coefficient of determination is 0.95, this means that 95% of the y values were predicted correctly by the regression line.

The value of the sum of squares for regression SSR can never be larger than the value of total sum of squares SST.

In a simple linear regression model, testing whether the slope $\beta$ ₁ of the population regression line could be zero is the same as testing whether or not the population coefficient of correlation $\rho$ equals zero.

A zero correlation coefficient between a pair of random variables means that there is no linear relationship between the random variables.

If the coefficient of determination is 1.0, then the coefficient of correlation must be 1.0.

The probability distribution of the error variable $\varepsilon$ is normal, with mean E( $\varepsilon$ ) = 0, and standard deviation $\sigma$ _{$\varepsilon$}=1.

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0.

If the coefficient of correlation is -0.81, then the percentage of the variation in y that is explained by the regression line is 81%.

When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same, the standard error of estimate s_{$\varepsilon$} will be 0.0.

A zero population correlation coefficient for x and y means that there is no type of relationship whatsoever between x and y.

Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y.

If there is no linear relationship between two variables x and y, the coefficient of determination must be -1.0.

If the coefficient of determination is 0.95, this means that 95% of the variation in the independent variable x can be explained by the y variable.

The coefficient of determination is equal to the coefficient of correlation squared.

In a simple linear regression problem, the least squares line is $\tilde { y } = - 3.75 + 1.25 x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be -0.90.

In simple linear regression, the denominator of the standard error of estimate s_{$\varepsilon$} is $\sqrt { n - 2 }$ .

Quiz 16: Simple Linear Regression and Correlation

If the error variable ε\varepsilonε is normally distributed, the test statistic for testing H0: β\betaβ 1 = 0 has a Student t-distribution with n -2 degrees of freedom.

If the value of the sum of squares for error SSE equals zero, then the coefficient of determination must equal zero.

The value of the sum of squares for regression SSR can never be larger than the value of sum of squares for error SSE.

If the coefficient of determination is 0.95, this means that 95% of the y values were predicted correctly by the regression line.

The value of the sum of squares for regression SSR can never be larger than the value of total sum of squares SST.

In a simple linear regression model, testing whether the slope β\betaβ 1 of the population regression line could be zero is the same as testing whether or not the population coefficient of correlation ρ\rhoρ equals zero.

A zero correlation coefficient between a pair of random variables means that there is no linear relationship between the random variables.

If the coefficient of determination is 1.0, then the coefficient of correlation must be 1.0.

The probability distribution of the error variable ε\varepsilonε is normal, with mean E( ε\varepsilonε ) = 0, and standard deviation σ\sigmaσ ε\varepsilonε =1.

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0.

If the coefficient of correlation is -0.81, then the percentage of the variation in y that is explained by the regression line is 81%.

When the actual values y of a dependent variable and the corresponding predicted values y^\hat { y }y^​ are the same, the standard error of estimate s ε\varepsilonε will be 0.0.

A zero population correlation coefficient for x and y means that there is no type of relationship whatsoever between x and y.

Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y.

If there is no linear relationship between two variables x and y, the coefficient of determination must be -1.0.

If the coefficient of determination is 0.95, this means that 95% of the variation in the independent variable x can be explained by the y variable.

The coefficient of determination is equal to the coefficient of correlation squared.

In a simple linear regression problem, the least squares line is y~=−3.75+1.25x\tilde { y } = - 3.75 + 1.25 xy~​=−3.75+1.25x , and the coefficient of determination is 0.81. The coefficient of correlation must be -0.90.

In simple linear regression, the denominator of the standard error of estimate s ε\varepsilonε is n−2\sqrt { n - 2 }n−2​ .

If the error variable $\varepsilon$ is normally distributed, the test statistic for testing H₀: $\beta$ ₁ = 0 has a Student t-distribution with n -2 degrees of freedom.

In a simple linear regression model, testing whether the slope $\beta$ ₁ of the population regression line could be zero is the same as testing whether or not the population coefficient of correlation $\rho$ equals zero.

The probability distribution of the error variable $\varepsilon$ is normal, with mean E( $\varepsilon$ ) = 0, and standard deviation $\sigma$ _{$\varepsilon$}=1.

When the actual values y of a dependent variable and the corresponding predicted values $\hat { y }$ are the same, the standard error of estimate s_{$\varepsilon$} will be 0.0.

In a simple linear regression problem, the least squares line is $\tilde { y } = - 3.75 + 1.25 x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be -0.90.

In simple linear regression, the denominator of the standard error of estimate s_{$\varepsilon$} is $\sqrt { n - 2 }$ .