Question 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%.

Accepted Answer

The percentage of the variation in y that is explained by the regression line is determined by the coefficient of determination, which is the square of the correlation coefficient. Thus, (-0.81)^2 = 0.6561 or 65.61%, not 81%.

Question 2

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

A prediction interval is indeed used to predict a one-time occurrence for a particular value of y when the independent variable is a given x value. It gives a range of values within which we can expect the outcome to lie with a certain level of confidence or probability.

Question 3

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

Question 4

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 5

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 6

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

Accepted Answer

The coefficient of determination (R^2) being 1.0 means that the model explains all the variability of the response data around its mean. However, the coefficient of correlation (R) could be either 1.0 or -1.0, indicating a perfect positive or negative linear relationship, respectively.

Question 7

If the error variable e is normally distributed, the test statistic for testing H ₀: b ₁ = 0 has a Student t -distribution with n - 2 degrees of freedom.

Accepted Answer

When the error term (e) in a regression model is normally distributed, the test statistic for testing the null hypothesis that the slope coefficient (b1) equals zero (H0: b1 = 0) indeed follows a Student's t-distribution with n - 2 degrees of freedom, where n is the number of observations. This is because the estimate of the slope coefficient is normally distributed if the errors are normally distributed, and the t-distribution is used when estimating the standard deviation of the coefficient from the sample.

Question 8

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 variance in y that can be explained by the linear relationship with x. It can range from 0 to 1, where 0 means no linear relationship and 1 means perfect linear relationship. Therefore, if there is no linear relationship between x and y, the coefficient of determination will be 0, not -1.

Question 9

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) represents the proportion of the variance in the dependent variable (y) that is explained by the independent variable(s) in the regression model. Therefore, if R-squared is 0.95, it means that 95% of the variance in the dependent variable is explained by the independent variable(s), not that 95% of the individual y values were predicted correctly by the regression line.

Question 10

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 examines the linear relationship between two variables.

Question 11

The probability distribution of the error variable e is normal, with mean E ( e )= 0, and standard deviation s _e =1.

Accepted Answer

The statement is incomplete and lacks context. The probability distribution of the error variable e being normal with mean 0 and standard deviation 1 is a common assumption in statistics, but it is not universally true without specifying the context or model.

Question 12

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 variables. However, there could still be a non-linear relationship or other types of relationships between the variables.

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_e will be 0.0.

Accepted Answer

The answer of When the actual values y of a...

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

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

The answer of In a simple linear regression problem, the...

Question 16

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 statement is true. The total sum of squares SST is the sum of the squared deviations of the dependent variable around its mean, while the sum of squares for regression SSR is the sum of the squared deviations of the predicted values around the mean. Since the predicted values are based on the regression equation, which is chosen to minimize the sum of squared errors, SSR cannot be larger than SST.

Question 17

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

Accepted Answer

This is a true statement. The formula for the coefficient of determination (R-squared) is the square of the coefficient of correlation (r).

Question 18

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 SSE equals zero, it means that the regression line perfectly fits the data points and the coefficient of determination (R-squared) equals 1, indicating that all of the variation in the dependent variable is explained by the independent variable.

Question 19

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

Accepted Answer

In simple linear regression, testing whether the slope (β1) is zero is equivalent to testing if there is no linear relationship between the variables, which is the same as testing if the population correlation coefficient (r) is zero. If the slope is zero, it implies that changes in the predictor variable do not predict changes in the response variable, indicating no linear correlation.

Question 20

In simple linear regression, the denominator of the standard error of estimate s_e is .

Accepted Answer

The answer of In simple linear regression, the denominator of...

Quiz 16: Simple Linear Regression and Correlat