91) True or False: The Regression Sum of Squares (SSR) can never be greater than the Total Sum of Squares (SST).
92) True or False: The coefficient of determination represents the ratio of SSR to SST.
93) True or False: Regression analysis is used for prediction, while correlation analysis is used to measure the strength of the association between two numerical variables.
94) True or False: The value of r is always positive.
95) 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.
96) 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.
97) 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
98) True or False: Data that exhibit an autocorrelation effect violate the regression assumption of independence.
99) True or False: The Durbin-Watson D statistic is used to check the assumption of normality.
100) 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 ________.
The Correct Answer and Explanation is :
Let’s go through each statement and question with the correct answers and explanations:
91) True or False: The Regression Sum of Squares (SSR) can never be greater than the Total Sum of Squares (SST).
- True
Explanation: The Total Sum of Squares (SST) is the total variation in the dependent variable (Y) from its mean, while the Regression Sum of Squares (SSR) is the part of that variation explained by the regression model. SSR can never exceed SST because the regression model cannot explain more variation than exists in the data.
92) True or False: The coefficient of determination represents the ratio of SSR to SST.
- True
Explanation: The coefficient of determination (R²) is indeed the ratio of SSR to SST. It measures the proportion of the total variation in the dependent variable that is explained by the independent variables in the model.
93) True or False: Regression analysis is used for prediction, while correlation analysis is used to measure the strength of the association between two numerical variables.
- True
Explanation: Regression analysis aims to model the relationship between a dependent and one or more independent variables, with a focus on prediction. Correlation analysis, on the other hand, measures the strength and direction of a linear relationship between two numerical variables but does not establish causation or predict one variable based on the other.
94) True or False: The value of r is always positive.
- False
Explanation: The correlation coefficient (r) can be positive or negative, depending on the direction of the linear relationship between the variables. A positive r indicates a positive relationship, while a negative r indicates an inverse relationship.
95) In performing a regression analysis involving two numerical variables, you are assuming:
- B) the variation around the line of regression is the same for each X value.
Explanation: This is an assumption of homoscedasticity, meaning the variance of errors is constant across all levels of the independent variable (X). The other options (A and C) are incorrect assumptions in basic regression analysis.
96) Which of the following assumptions concerning the probability distribution of the random error term is stated incorrectly?
- C) The variance of the distribution increases as X increases.
Explanation: This would violate the assumption of homoscedasticity, which assumes that the variance of the error term is constant across all levels of X (the independent variable). A heteroscedastic error term would have varying variance with different X values.
97) Based on the residual plot below, you will conclude that there might be a violation of which of the following assumptions?
- C) Homoscedasticity
Explanation: A residual plot that shows a pattern (e.g., a funnel shape) suggests heteroscedasticity, indicating that the variance of the errors changes as the independent variable (X) changes, violating the assumption of homoscedasticity.
98) True or False: Data that exhibit an autocorrelation effect violate the regression assumption of independence.
- True
Explanation: Autocorrelation occurs when the residuals (errors) from one observation are correlated with the residuals from another observation, which violates the assumption that errors are independent of each other in regression analysis.
99) True or False: The Durbin-Watson D statistic is used to check the assumption of normality.
- False
Explanation: The Durbin-Watson statistic tests for autocorrelation in the residuals (the assumption of independence), not for normality. To test normality, other tests like the Shapiro-Wilk test or visualizations like histograms or Q-Q plots are used.
100) 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 __.
- 2
Explanation: The Durbin-Watson statistic is used to detect autocorrelation in the residuals. A value near 2 indicates no autocorrelation in the residuals. A value much smaller than 2 suggests positive autocorrelation, and a value much larger than 2 suggests negative autocorrelation.
These answers and explanations reflect core concepts in regression analysis, including assumptions, statistical tests, and interpretation of key measures such as R², Durbin-Watson, and residuals.