Question 1

If the probability plot of residuals resembles a straight line,the residuals show a fairly good fit to the normal distribution.

Accepted Answer

If the probability plot of residuals resembles a straight line, it indicates that the residuals follow a normal distribution. This is because a probability plot is a graphical tool used to assess the normality of a distribution. Therefore, option A is true.

Question 2

The F statistic in a multiple regression is significant if at least one of the predictors has a significant t statistic at a given α.

Accepted Answer

The F statistic tests the overall significance of the multiple regression model, while the t statistic tests the individual significance of each predictor variable. Therefore, if the F statistic is significant, at least one of the predictors must have a significant t statistic.

Question 3

If R2 and R2adj differ greatly,we should probably add a few predictors to improve the fit.

Accepted Answer

A large difference between R² and adjusted R² suggests that some predictors may not be contributing to the model, and adding more predictors could worsen the model by increasing complexity without improving fit.

Question 4

Confidence intervals for Y may be unreliable when the residuals are not normally distributed.

Accepted Answer

Confidence intervals are based on the assumption that the residuals (the differences between the observed data and the predicted values) are normally distributed. If the residuals are not normally distributed, the confidence intervals may not accurately reflect the true variability of the data and may therefore be unreliable.

Question 5

A common misinterpretation of the principle of Occam's Razor is that a simple regression model (rather than a multiple regression model)is always best.

Accepted Answer

Occam's Razor suggests that among competing hypotheses that predict equally well, the one with the fewest assumptions should be selected. However, this principle is often misinterpreted to mean that simpler models are always better, which is not necessarily true, especially in cases where multiple variables genuinely affect the outcome.

Question 6

R2adj can exceed R2 if there are several weak predictors.

Accepted Answer

Adjusted R-squared (R2adj) accounts for the number of predictors in the model and can never exceed the regular R-squared (R2); it adjusts downward when adding predictors that do not improve the model significantly.

Question 7

In a regression,the model with the best fit is preferred over all other models.

Accepted Answer

A model with the best fit might overfit the data, capturing noise rather than the underlying trend, which can lead to poor predictive performance on new, unseen data. It's important to balance fit with the model's complexity to ensure generalizability.

Question 8

For a certain firm,the regression equation Bonus = 2,000 + 257 Experience + 0.046 Salary describes employee bonuses with a standard error of 125.John has 10 years' experience,earns $50,000,and earned a bonus of $7,000.John is an outlier.

Accepted Answer

To determine if John is an outlier, we calculate his predicted bonus using the regression equation: Bonus = 2,000 + 257(10) + 0.046(50,000) = 2,000 + 2,570 + 2,300 = 6,870. John's actual bonus is $7,000, which is close to the predicted value, and the difference is not significantly larger than the standard error of 125, suggesting he is not an outlier.

Question 9

A binary (categorical)predictor should not be used along with nonbinary (numerical)predictors.

Accepted Answer

A binary predictor and nonbinary predictors can be used together in a model, as long as the necessary transformations and scaling are done to the nonbinary predictors.

Question 10

A negative estimated coefficient in a regression usually indicates a weak predictor.

Accepted Answer

A negative estimated coefficient in a regression indicates a negative relationship between the predictor variable and the response variable, but it does not necessarily mean that it is a weak predictor. The strength of a predictor is determined by its statistical significance, magnitude, and practical significance in relation to the research question.

Question 11

In a multiple regression with 3 predictors in a sample of 25 U.S.cities,we would use F3,21 in a test of overall significance.

Accepted Answer

In a multiple regression with 3 predictors and a sample of 25 observations, we would have 21 degrees of freedom for error. The test of overall significance for the regression model would be calculated using an F-distribution with 3 degrees of freedom for the numerator (numerator degrees of freedom = number of predictors) and 21 degrees of freedom for the denominator (denominator degrees of freedom = sample size minus number of predictors minus 1), resulting in an F3,21 distribution.

Question 12

If a regression model's F test statistic is Fcalc = 43.82,we could say that the explained variance is approximately 44 percent.

Accepted Answer

The F test statistic is used to test the overall significance of a regression model. It is not directly related to the percentage of variability in the dependent variable that is explained by the independent variable(s) in the model. Therefore, we cannot make any statement about the explained variance based solely on the F test statistic.

Question 13

The model Y = β0 + β1X + β2X2 cannot be estimated by Excel because of the nonlinear term.

Accepted Answer

Excel can estimate models with nonlinear terms like X^2 by using the regression tool, treating X^2 as an additional independent variable.

Question 14

The effect of a binary predictor is to shift the regression intercept.

Accepted Answer

A binary predictor, also known as a dummy variable, can shift the regression intercept because it represents a different category or group within the predictor variable. For example, if the binary predictor represents gender, there will be separate intercepts for males and females, which will shift the overall intercept of the regression model.

Question 15

Evans' Rule says that if n = 50 you need at least 5 predictors to have a good model.

Accepted Answer

Evans' Rule states that as a general guideline, the number of predictors should be greater than 5 times the number of observations (n) for a good model. However, this is not a hard and fast rule and the optimal number of predictors can vary depending on the dataset and the purpose of the model. Therefore, it is not accurate to say that if n = 50, you need at least 5 predictors for a good model.

Question 16

In regression,the dependent variable is referred to as the response variable.

Accepted Answer

In regression, the variable that is being predicted or estimated is referred to as the dependent or response variable.

Question 17

There is one residual for each predictor in the regression model.

Accepted Answer

There is one residual for each observation in the regression model, not for each predictor. Residuals are the differences between the observed values and the values predicted by the model.

Question 18

A predictor whose pairwise correlation with Y is near zero can still have a significant t-value in a multiple regression when other predictors are included.

Accepted Answer

This is because in multiple regression, predictors can have a significant effect on the outcome variable even if their individual correlation with the outcome variable is low. This is because the overall relationship between the predictors and the outcome variable is measured, taking into account the effects of all predictors included in the model.

Question 19

The random error term in a regression model reflects all factors omitted from the model.

Accepted Answer

The random error term in a regression model represents the unpredictable variation in the dependent variable that remains after accounting for the factors included in the model. It may be influenced by factors that were not included in the model, but it also includes other sources of variation, such as measurement error or natural variability.

Question 20

A parsimonious model is one with many weak predictors but a few strong ones.

Accepted Answer

A parsimonious model is one with few predictors, both weak and strong, that are sufficient to explain the outcome variable.

Quiz 13: Multiple Regression