Question 1

If R² and R²_adj differ greatly, we should probably add a few predictors to improve the fit.

Accepted Answer

The two models mentioned are identical in terms of predictors used (RS, U, P, S, B) and their configurations, with the difference being the addition of "adj" in the second model, which likely refers to an adjustment or interaction term rather than a new predictor. Adding more predictors is not suggested by a difference between these models, but rather exploring interaction effects or model specifications.

Question 2

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 3

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.

Question 4

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 5

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 6

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 can happen because the presence of other predictors can control for any confounding effects and allow the predictor with low pairwise correlation to show its unique contribution to the outcome variable.

Question 7

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 using tools like Solver or the Analysis ToolPak, which support nonlinear regression analysis.

Question 8

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

Accepted Answer

The F test statistic in regression analysis measures the overall significance of the model, not the percentage of variance explained. The percentage of variance explained is indicated by the R-squared value.

Question 9

In a multiple regression with 3 predictors in a sample of 25 U.S. cities, we would use F_3,21 in a test of overall significance.

Accepted Answer

F_3,21 is the F-statistic used in the test of overall significance in a multiple regression with 3 predictors. Therefore, the statement is true.

Question 10

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

Accepted Answer

A negative estimated coefficient in a regression does not necessarily indicate a weak predictor. It simply means that as the predictor variable increases, the response variable tends to decrease. The strength of the predictor is determined by the magnitude and statistical significance of the coefficient, not just its sign.

Question 11

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

Accepted Answer

The random error term reflects factors that are not accounted for in the model, including measurement errors, unmeasured variables, and unpredictable variation. It is often referred to as the "noise" in the data that cannot be explained by the model's independent variables.

Question 12

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

Accepted Answer

A binary predictor can be thought of as a variable that takes on only two values (e.g. 0 or 1). When included in a regression model, the effect of the binary predictor is to add a fixed amount to the intercept for one group and subtract the same amount from the intercept for the other group. This shift in intercept can be interpreted as a change in the baseline level of the outcome variable for the two groups defined by the binary predictor.

Question 13

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

Accepted Answer

True, the probability plot of residuals is a graphical way to assess whether the residuals of a statistical model follow a normal distribution. If the plot looks like a straight line, it suggests that the residuals have a reasonably good fit to a normal distribution. If the plot is curved or nonlinear, it suggests that the residuals do not follow a normal distribution and may indicate a problem with the model.

Question 14

R²_adj can exceed R² if there are several weak predictors.

Accepted Answer

RS1U1P12S1S1P0S1U1B1adjS1U1U1B0 is designed to adjust for the number of predictors in a model, penalizing for adding weak predictors, thus it cannot exceed RS1U1P12S1S1P0 when weak predictors are added; instead, it should decrease or stay the same to reflect the penalty for model complexity.

Question 15

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

Accepted Answer

It is common and valid to use a binary predictor alongside nonbinary predictors in many statistical models. For example, in logistic regression, a binary predictor (also known as a dummy variable or indicator variable) is often used alongside other numerical predictors to predict the probability of an outcome.

Question 16

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 determines whether at least one of the predictors in the model is contributing significantly to the variation in the dependent variable. If the F statistic is significant, it means that the overall model is contributing significantly to the variation in the dependent variable, and at least one predictor is significant. The significant t statistic for any of the predictors indicates that the particular predictor is contributing significantly to the variation in the dependent variable, but it is not necessary for every predictor to have a significant t statistic for the F statistic to be significant.

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

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

Accepted Answer

Confidence intervals are calculated based on the assumption that the residuals (the differences between the observed values of Y and the predicted values from the regression model) are normally distributed. If this assumption is not met, the confidence intervals may not accurately capture the true range of values for Y, leading to unreliable results.

Question 19

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 20

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

Accepted Answer

This statement is true. In regression analysis, the response variable is the variable that is being predicted or modeled. It is also known as the dependent variable because its value depends on the values of the predictor or independent variables.

Quiz 13: Multiple Regression

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