Question 1

The standard error of estimate, $$S _  { \varepsilon }$$ , is given by:

Accepted Answer

The standard error of estimate is defined as the standard deviation of the residuals. Therefore, it is the square root of SSE divided by the degrees of freedom, which is (n-2) for simple linear regression. Hence, the formula for the standard error of estimate is given by option C, i.e., $$\sqrt { SSE/(n-2) }$$.

Question 2

Which of the following best describes the coefficient of determination?

Accepted Answer

The coefficient of determination, denoted as R squared, measures the proportion of variation in the dependent variable (y) that can be explained by the independent variable (x) in a linear regression model. Therefore, choice C accurately describes the coefficient of determination. Choices A and D are partially true, as they relate to the percentage of variation explained by the model, but they refer to the wrong variable. Choice B is incorrect as it does not refer to the coefficient of determination.

Question 3

If the coefficient of determination is 81%, and the linear regression model has a negative slope, what is the value of the coefficient of correlation?

Accepted Answer

The coefficient of correlation, denoted by "r," is the square root of the coefficient of determination (r = sqrt(R2)). Since the coefficient of determination is 81%, then the coefficient of correlation is sqrt(0.81) = 0.9. Additionally, the fact that the linear regression model has a negative slope indicates that there is a negative correlation between the variables, which is reflected in the negative sign of the slope. Therefore, the best choice is D, −0.90.

Question 4

Which of the following techniques is used to predict the value of one variable on the basis of other variables?

Accepted Answer

Regression analysis is used to predict the value of one variable (dependent variable) on the basis of other variables (independent variables). Correlation analysis (option A) is used to determine the relationship between two variables but not to predict values. Coefficient of correlation (option B) is a measure of the strength and direction of the relationship between two variables. Covariance (option C) measures how two variables vary together but is not commonly used for prediction.

Question 5

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line 
 = 80 000 + 5x. This implies that an:

Accepted Answer

The answer of A regression analysis between sales (in $1000)...

Question 6

The standardised residual is defined as:

Accepted Answer

The formula for calculating the standardised residual is dividing the residual by the standard error of estimate. This helps to standardize the residuals for easier comparison and interpretation.

Question 7

When all the actual values of y and the predicted values of y are equal, the standard error of estimate will be:

Accepted Answer

The standard error of estimate measures the variability of actual y values around the predicted y values. If all the actual and predicted values are equal, there is no variability, and therefore the standard error of estimate is 0.

Question 8

The standard error of estimate, $$S _  { \varepsilon }$$ , is a measure of:

Accepted Answer

The standard error of estimate, $$S _  { \varepsilon }$$ , measures the variation of y (the response variable) around the regression line. It is a representation of the scatter of the data points around the estimated regression line.

Question 9

In regression analysis, if the coefficient of correlation is -1.0, then:

Accepted Answer

If the coefficient of correlation is -1.0, it means there is a perfect negative linear relationship between the independent and dependent variables. Therefore, the sum of squares for regression and total variation in y (which measures the total variability of the dependent variable) would be equal.

Question 10

A regression analysis between height y (in cm) and age x (in years) of 2 to 10 years old boys yielded the least squares line y-hat = 87 + 6.5x. This implies that by each additional year height is expected to:

Accepted Answer

According to the least squares line, for each additional year of age (x), the height (y) is expected to increase by 6.5cm. Therefore, choice B is the correct answer.

Question 11

Which of the following statements best describes why a linear regression is also called a least squares regression model?

Accepted Answer

A linear regression is called a least squares regression model because it calculates the regression line by minimizing the sum of the square of the differences between each actual y data value and the predicted y value. This means that the line of best fit is the one that minimizes the overall error between the predicted values and the actual values.

Question 12

Given the data points (x,y) = (3,3), (4,4), (5,5), (6,6), (7,7), the least squares estimates of the y-intercept and slope are respectively:

Accepted Answer

The data points lie perfectly on the line y = x, which has a slope of 1 and a y-intercept of 0.

Question 13

In testing the hypotheses: $$H _ { 0 } : \rho _ { s } = 0$$ $$H _ { 1 } : D _ { s } \neq 0$$ , the Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

Accepted Answer

The test statistic for Spearman's rank correlation is calculated using the formula: $ t = \frac{r_s \sqrt{n-2}}{\sqrt{1-r_s^2}} $. Substituting $ r_s = 0.389 $ and $ n = 50 $ gives approximately 2.723.

Question 14

In order to estimate with 95% confidence the expected value of y in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

Accepted Answer

Since we are estimating the expected value of y in a simple linear regression, we use the t-distribution with n-2 degrees of freedom. In this case, n=10, so there are 8 degrees of freedom. We need to find the t-value that corresponds to a 95% confidence level and 8 degrees of freedom. Looking at a t-table, we see that the closest value is 2.306, which corresponds to a two-tailed probability of 0.05 and 8 degrees of freedom. Therefore, the answer is B.

Question 15

Which of the following best describes if we want to test for a linear relationship between x and y, in regression analysis?

Accepted Answer

The answer of Which of the following best describes if...

Question 16

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line 
 = 75 +6x. This implies that if $800 is spent on advertising, then the predicted amount of sales (in dollars) is:

Accepted Answer

The answer of A regression analysis between sales (in $1000)...

Question 17

In the first-order linear regression model, the population parameters of the y-intercept and the slope are:

Accepted Answer

The population parameters in the first-order linear regression model are the y-intercept ($$\beta _ { 0 }$$) and the slope ($$\beta _ { 1 }$$). Option D is the only choice that correctly identifies both population parameters.

Question 18

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?

Accepted Answer

All of these statistics and procedures can be used to determine whether a linear model should be employed. 
The standard error of estimate measures the variability of the residuals and indicates how well the model fits the data. 
The coefficient of determination represents the proportion of variation in the dependent variable that is explained by the independent variable(s) in the model. A higher value suggests a better fit. 
The t-test of the slope tests the hypothesis that the slope of the regression line is equal to zero, indicating no relationship between the variables. If the p-value is less than the significance level, it suggests a significant relationship and supports the use of a linear model.

Question 19

Which of the following statements is correct when all the actual values of y are on an upward sloping regression line?

Accepted Answer

When all the actual values of y are on an upward sloping regression line, it indicates a positive relationship between x and y. In such a case, the coefficient of correlation (r) will be equal to 1 and the sign of both the coefficient of correlation and the slope will be positive. Therefore, option C is the correct statement. Options A, B, and D are incorrect.

Question 20

If the coefficient of correlation between x and y is close to −1.0, which of the following statements is correct?

Accepted Answer

A coefficient of correlation of −1.0 indicates a strong, negative linear relationship between x and y. However, this does not necessarily imply a causal relationship between x and y, as correlation does not imply causation. Therefore, option D is the correct answer.

Quiz 18: Simple Linear Regression and Correlation

The standard error of estimate, $S _ { \varepsilon }$ , is given by:

Which of the following best describes the coefficient of determination?

If the coefficient of determination is 81%, and the linear regression model has a negative slope, what is the value of the coefficient of correlation?

Which of the following techniques is used to predict the value of one variable on the basis of other variables?

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line = 80 000 + 5x. This implies that an:

The standardised residual is defined as:

When all the actual values of y and the predicted values of y are equal, the standard error of estimate will be:

The standard error of estimate, $S _ { \varepsilon }$ , is a measure of:

In regression analysis, if the coefficient of correlation is -1.0, then:

A regression analysis between height y (in cm) and age x (in years) of 2 to 10 years old boys yielded the least squares line y-hat = 87 + 6.5x. This implies that by each additional year height is expected to:

Which of the following statements best describes why a linear regression is also called a least squares regression model?

Given the data points (x,y) = (3,3) , (4,4) , (5,5) , (6,6) , (7,7) , the least squares estimates of the y-intercept and slope are respectively:

In testing the hypotheses: $H _ { 0 } : \rho _ { s } = 0$ $H _ { 1 } : D _ { s } \neq 0$ , the Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

In order to estimate with 95% confidence the expected value of y in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

Which of the following best describes if we want to test for a linear relationship between x and y, in regression analysis?

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line = 75 +6x. This implies that if $800 is spent on advertising, then the predicted amount of sales (in dollars) is:

In the first-order linear regression model, the population parameters of the y-intercept and the slope are:

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?

Which of the following statements is correct when all the actual values of y are on an upward sloping regression line?

If the coefficient of correlation between x and y is close to −1.0, which of the following statements is correct?

Quiz 18: Simple Linear Regression and Correlation

The standard error of estimate, SεS _ { \varepsilon }Sε​ , is given by:

Which of the following best describes the coefficient of determination?

If the coefficient of determination is 81%, and the linear regression model has a negative slope, what is the value of the coefficient of correlation?

Which of the following techniques is used to predict the value of one variable on the basis of other variables?

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line = 80 000 + 5x. This implies that an:

The standardised residual is defined as:

When all the actual values of y and the predicted values of y are equal, the standard error of estimate will be:

The standard error of estimate, SεS _ { \varepsilon }Sε​ , is a measure of:

In regression analysis, if the coefficient of correlation is -1.0, then:

A regression analysis between height y (in cm) and age x (in years) of 2 to 10 years old boys yielded the least squares line y-hat = 87 + 6.5x. This implies that by each additional year height is expected to:

Which of the following statements best describes why a linear regression is also called a least squares regression model?

Given the data points (x,y) = (3,3) , (4,4) , (5,5) , (6,6) , (7,7) , the least squares estimates of the y-intercept and slope are respectively:

In testing the hypotheses: H0:ρs=0H _ { 0 } : \rho _ { s } = 0H0​:ρs​=0 H1:Ds≠0H _ { 1 } : D _ { s } \neq 0H1​:Ds​=0 , the Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

In order to estimate with 95% confidence the expected value of y in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

Which of the following best describes if we want to test for a linear relationship between x and y, in regression analysis?

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line = 75 +6x. This implies that if $800 is spent on advertising, then the predicted amount of sales (in dollars) is:

In the first-order linear regression model, the population parameters of the y-intercept and the slope are:

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?

Which of the following statements is correct when all the actual values of y are on an upward sloping regression line?

If the coefficient of correlation between x and y is close to −1.0, which of the following statements is correct?

The standard error of estimate, $S _ { \varepsilon }$ , is given by:

The standard error of estimate, $S _ { \varepsilon }$ , is a measure of:

In testing the hypotheses: $H _ { 0 } : \rho _ { s } = 0$ $H _ { 1 } : D _ { s } \neq 0$ , the Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is: