Question 1

Based on the residual plot below, you will conclude that there might be a violation of which of the following assumptions. 
A) Linearity of the relationship
B) Normality of errors
C) Homoscedasticity
D) Independence of errors

Accepted Answer

The answer of Based on the residual plot below, you...

Question 2

SCENARIO 13-6
The following Excel tables are obtained when "Score received on an exam (measured in percentage
points)" (Y) is regressed on "percentage attendance" (X) for 22 students in a Statistics for Business
and Economics course. $$\begin{array}{l}
\begin{array} { l r } 
\hline  { \text { Regression Statistics } } \
\hline \text { Multiple R } & 0.142620229 \
\text { R Square } & 0.02034053 \
\text { Standard Error } & 20.25979924 \
\text { Observations } & 22 \
\hline
\end{array}\\
\begin{array} { l r r r l } 
\hline & \text { Coefficients } & \text { Standard Error } & \text { T Stat } & P \text {-value } \
\hline \text { Intercept } & 39.39027309 & 37.24347659 & 1.057642216 & 0.302826622 \
\text { Attendance } & 0.340583573 & 0.52852452 & 0.644404489 & 0.526635689 \
\hline
\end{array}
\end{array}$$
-Referring to Scenario 13-6, which of the following statements is true?
A) If attendance increases by 0.341%, the estimated mean score received will increase by 1 percentage point.
B) If attendance increases by 1%, the estimated mean score received will increase by 39.39 percentage points.
C) If attendance increases by 1%, the estimated mean score received will increase by 0.341 percentage points.
D) If the score received increases by 39.39%, the estimated mean attendance will go up by 1%.

Accepted Answer

According to the regression results, the estimated regression coefficient for X (percentage attendance) is 0.341. This means that, on average, for each 1% increase in attendance, the estimated mean score received will increase by 0.341 percentage points. Therefore, statement C is true, while A, B, and D are false.

Question 3

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.

Accepted Answer

The assumption that the variance of the distribution increases as X increases is incorrect. The correct assumption is that the variance of the distribution of the random error term is constant for all values of X.

Question 4

If the residuals in a regression analysis of time-ordered data are not correlated, the value of the
Durbin-Watson D statistic should be near __________.

Accepted Answer

The answer of If the residuals in a regression analysis...

Question 5

If the Durbin-Watson statistic has a value close to 4, which assumption is violated?
A) Normality of the errors.
B) Independence of errors.
C) Homoscedasticity.
D) None of the above.

Accepted Answer

The Durbin-Watson statistic tests for autocorrelation, meaning whether there is a pattern in the residuals over time. The statistic ranges from 0 to 4, with a value of 2 indicating no autocorrelation. If the value is close to 4, it indicates negative autocorrelation, meaning that residuals at one time point are negatively correlated with residuals at the next time point. This violates the assumption of independence of the errors.

Question 6

The Regression Sum of Squares (SSR) can never be greater than the Total Sum of
Squares (SST).

Accepted Answer

This is true because SSR is the sum of the squared differences between the predicted values and the mean of the dependent variable, whereas SST is the sum of the squared differences between the actual values and the mean of the dependent variable. Since the predicted values are based on the relationship between the independent and dependent variables, they will always be closer to the actual values than the mean, leading to a smaller sum of squares for SSR compared to SST.

Question 7

The residuals represent
A) the difference between the actual Y values and the mean of Y.
B) the difference between the actual Y values and the predicted Y values.
C) the square root of the slope.
D) the predicted value of Y for the average X value.

Accepted Answer

Residuals are the difference between the actual Y values and the predicted Y values. The other options are incorrect. The mean of Y is not involved in calculating residuals, the square root of the slope is not a measure of residuals, and the predicted value of Y for the average X value is not the same as the residuals.

Question 8

SCENARIO 13-6
The following Excel tables are obtained when "Score received on an exam (measured in percentage
points)" (Y) is regressed on "percentage attendance" (X) for 22 students in a Statistics for Business
and Economics course. $$\begin{array}{l}
\begin{array} { l r } 
\hline  { \text { Regression Statistics } } \
\hline \text { Multiple R } & 0.142620229 \
\text { R Square } & 0.02034053 \
\text { Standard Error } & 20.25979924 \
\text { Observations } & 22 \
\hline
\end{array}\\
\begin{array} { l r r r l } 
\hline & \text { Coefficients } & \text { Standard Error } & \text { T Stat } & P \text {-value } \
\hline \text { Intercept } & 39.39027309 & 37.24347659 & 1.057642216 & 0.302826622 \
\text { Attendance } & 0.340583573 & 0.52852452 & 0.644404489 & 0.526635689 \
\hline
\end{array}
\end{array}$$
-Referring to Scenario 13-6, which of the following statements is true?
A) 14.26% of the total variability in score received can be explained by percentage attendance.
B) 14.2% of the total variability in percentage attendance can be explained by score received.
C) 2% of the total variability in score received can be explained by percentage attendance.
D) 2% of the total variability in percentage attendance can be explained by score received.

Accepted Answer

The answer of SCENARIO 13-6
The following Excel tables are obtained...

Question 9

SCENARIO 13-5
The managing partner of an advertising agency believes that his company's sales are related to the
industry sales. He uses Microsoft Excel to analyze the last 4 years of quarterly data with
the following results: 
-Referring to Scenario 13-5, the partner wants to test for autocorrelation using the Durbin-Watson
statistic. Using a level of significance of 0.05, the critical values of the test are dL = ________,
and dU = ________.

Accepted Answer

The answer of SCENARIO 13-5
The managing partner of an advertising...

Question 10

If the plot of the residuals is fan shaped, which assumption is violated?
A) Normality.
B) Homoscedasticity.
C) Independence of errors.
D) No assumptions are violated, the graph should resemble a fan.

Accepted Answer

A fan-shaped plot of residuals indicates a violation of homoscedasticity assumption, which means that the variances of the error terms are not constant across the range of predicted values.

Question 11

Regression analysis is used for prediction, while correlation analysis is used to
measure the strength of the association between two numerical variables.

Accepted Answer

Regression analysis involves using one variable (the predictor or independent variable) to make predictions about another variable (the response or dependent variable). On the other hand, correlation analysis simply measures the strength and direction of the linear relationship between two variables, without making any predictions.

Question 12

Data that exhibit an autocorrelation effect violate the regression assumption of
independence.

Accepted Answer

Autocorrelation occurs when there is a correlation between a variable and its lagged versions. This violates the assumption of independence because it means that the value of the variable at one point in time is dependent on its value at a previous point in time. Therefore, the error terms will also be correlated and the regression assumption of independence is violated.

Question 13

If the Durbin-Watson statistic has a value close to 0, which assumption is violated?
A) Normality of the errors.
B) Independence of errors.
C) Homoscedasticity.
D) None of the above.

Accepted Answer

The Durbin-Watson statistic tests for independence of errors in a regression model. If the value is close to 0, it indicates positive autocorrelation, which means the errors are not independent. Therefore, assumption B (independence of errors) is violated.

Question 14

The coefficient of determination (r2) tells you
A) that the coefficient of correlation (r) is larger than 1.
B) whether r has any significance.
C) that you should not partition the total variation.
D) the proportion of total variation that is explained.

Accepted Answer

The coefficient of determination (r2) represents the proportion of total variation in the dependent variable that is explained by the independent variable(s). Therefore, option D is the correct answer. Option A is incorrect because the coefficient of correlation (r) cannot be larger than 1. Option B is incorrect because the significance of r is typically determined through hypothesis testing, not through the calculation of r2. Option C is incorrect because partitioning the total variation may be necessary or appropriate depending on the research question and design.

Question 15

The Durbin-Watson D statistic is used to check the assumption of normality.

Accepted Answer

The Durbin-Watson D statistic is used to check for autocorrelation (serial correlation) in the errors of a regression model, not for normality.

Question 16

The coefficient of determination represents the ratio of SSR to SST.

Accepted Answer

This statement is true. The coefficient of determination, often denoted as R-squared, is calculated as SSR/SST, where SSR is the sum of squares of the regression (explained variation) and SST is the total sum of squares (total variation). Therefore, the coefficient of determination represents the proportion of the total variation in the dependent variable that can be explained by the independent variable(s).

Question 17

The standard error of the estimate is a measure of
A) total variation of the Y variable.
B) the variation around the sample regression line.
C) explained variation.
D) the variation of the X variable.

Accepted Answer

The standard error of the estimate measures the variation or spread of the residuals around the sample regression line, which represents the deviation of actual Y values from predicted Y values.

Question 18

SCENARIO 13-5
The managing partner of an advertising agency believes that his company's sales are related to the
industry sales. He uses Microsoft Excel to analyze the last 4 years of quarterly data with
the following results: 
-Referring to Scenario 13-5, the partner wants to test for autocorrelation using the Durbin-Watson statistic. Using a level of significance of 0.05, the decision he should make is:
A) there is evidence of autocorrelation.
B) the test is unable to make a definite conclusion.
C) there is no evidence of autocorrelation.
D) there is not enough information to perform the test.

Accepted Answer

The Durbin-Watson statistic value is 1.905, which falls between the dL and dU values for a significance level of 0.05 and 8 observations. This indicates that there is no evidence of autocorrelation. Therefore, the managing partner should retain the assumption of no autocorrelation in the data.

Question 19

In performing a regression analysis involving two numerical variables, you are assuming
A) the variances of X and Y are equal.
B) the variation around the line of regression is the same for each X value.
C) that X and Y are independent.
D) All of the above.

Accepted Answer

In regression analysis, one of the key assumptions is that the variance of the residuals (errors) is constant across all levels of the independent variable (X). This is known as homoscedasticity, which is what choice B describes. Choices A and C are incorrect assumptions for regression analysis.

Question 20

The value of r is always positive.

Accepted Answer

The value of r, when referring to the correlation coefficient, can range from -1 to 1, indicating both positive and negative relationships between variables.

Quiz 13: Simple Linear Regression

Based on the residual plot below, you will conclude that there might be a violation of which of the following assumptions.

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

If the residuals in a regression analysis of time-ordered data are not correlated, the value of the Durbin-Watson D statistic should be near __________.

If the Durbin-Watson statistic has a value close to 4, which assumption is violated?

The Regression Sum of Squares (SSR) can never be greater than the Total Sum of Squares (SST).

The residuals represent

If the plot of the residuals is fan shaped, which assumption is violated?

Regression analysis is used for prediction, while correlation analysis is used to measure the strength of the association between two numerical variables.

Data that exhibit an autocorrelation effect violate the regression assumption of independence.

If the Durbin-Watson statistic has a value close to 0, which assumption is violated?

The coefficient of determination (r2) tells you

The Durbin-Watson D statistic is used to check the assumption of normality.

The coefficient of determination represents the ratio of SSR to SST.

The standard error of the estimate is a measure of

In performing a regression analysis involving two numerical variables, you are assuming

The value of r is always positive.