Question 1

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

Accepted Answer

This statement is true. The coefficient of determination (R-squared) is calculated by squaring the coefficient of correlation (R). It represents the proportion of variance in the dependent variable that is explained by the independent variable(s).

Question 2

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 3

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 SSR is the portion of SST explained by the regression model, and the rest of the variability is captured by the residual sum of squares (SSE). Therefore, SSR + SSE = SST, and since SSE is always positive, SSR cannot be greater than SST.

Question 4

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 (R^2) measures the proportion of variability in y that can be explained by the linear relationship with x. If there is no linear relationship, R^2 will be close to zero, but it will not necessarily be exactly zero. Therefore, the statement in the question is false.

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

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 7

A prediction interval is used when we want to predict a one-time occurrence for a particular value of y when the independent variable is a given x value.

Accepted Answer

This is a correct statement. A prediction interval provides a range of values with a certain level of confidence within which a future observation is expected to fall, given a specific value of the independent variable.

Question 8

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

Accepted Answer

If SSE equals zero, it means that there is no error left in the model, indicating a perfect fit. This would result in the coefficient of determination being equal to 1 (or 100%).

Question 9

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 10

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 11

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) actually measures the proportion of variation in the dependent variable y that can be explained by the independent variable x. So 0.95 means that 95% of the variation in y can be explained by x, not the other way around.

Question 12

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 statement is true because the population coefficient of correlation (ρ) between the two variables (F1 and S1) equals the slope (β) divided by the product of the standard deviations of the two variables. Therefore, if the slope is zero, the coefficient of correlation is also zero, and vice versa.

Question 13

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

Accepted Answer

When the actual values y of a dependent variable and the corresponding predicted values are the same, the residual error (also known as the residual or error term) ε will be 0. However, the standard error of estimate (which is a measure of the variability of the residuals around the regression line) is calculated by taking the square root of the sum of squared residuals divided by the degrees of freedom. Even if the residual error is 0, there will still be degrees of freedom (n - 2 for simple linear regression) and therefore a non-zero value for the standard error of estimate. So the statement is false.

Question 14

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 specifically looks for a linear relationship between two variables. If the relationship is not linear, other types of analysis may need to be used.

Question 15

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 there is no linear relationship between the two variables. However, it is possible that there may be a non-linear relationship or other types of relationships between the variables.

Question 16

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 17

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) measures the proportion of variability in the dependent variable (y) that is explained by the independent variable(s) in the regression model. An R-squared value of 0.95 means that 95% of the variability in y can be accounted for by the regression model, but it doesn't necessarily mean that 95% of the y values were predicted correctly by the regression line.

Question 18

In a simple linear regression problem, the least squares line is  , and the coefficient of determination is 0.81. The coefficient of correlation must be -0.90.

Accepted Answer

In a simple linear regression problem, the coefficient of correlation is equal to the square root of the coefficient of determination, so we have:

r = sqrt(R^2) = sqrt(0.81) = 0.9

Therefore, the statement is false, as the coefficient of correlation is 0.9, not F1F1F1S1F1F1F100.90.

Question 19

In simple linear regression, the denominator of the standard error of estimate s _{$\varepsilon$} is .

Accepted Answer

The denominator of the standard error of estimate in simple linear regression is the square root of the sum of squares of residuals divided by (n-2).

Question 20

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 the Student t-distribution for testing hypotheses involving the population mean when the population standard deviation is unknown and estimated from the sample. The test statistic follows a Student t-distribution with n-1 degrees of freedom, where n is the sample size. In this case, n is equal to the difference between the upper and lower index values of the error variable plus one, which is (10-1)+1 = 10. Therefore, the test statistic follows a Student t-distribution with 10-2 = 8 degrees of freedom.

Quiz 16: Simple Linear Regression and Correlation

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

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

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

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

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

A prediction interval is used when we want to predict a one-time occurrence for a particular value of y when the independent variable is a given x value.

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

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

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%.

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.

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.

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

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

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

In a simple linear regression problem, the least squares line is , 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 $In simple linear regression, the denominator of the standard error of estimate s <sub> \varepsilon </sub> is .$ .

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.

Quiz 16: Simple Linear Regression and Correlation

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

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

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

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

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

A prediction interval is used when we want to predict a one-time occurrence for a particular value of y when the independent variable is a given x value.

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

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

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.

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.

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

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

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

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

In a simple linear regression problem, the least squares line is , 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 .

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.

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

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.

In simple linear regression, the denominator of the standard error of estimate s _{$\varepsilon$} is $In simple linear regression, the denominator of the standard error of estimate s <sub> \varepsilon </sub> is .$ .

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.