Business Statistics Quiz: Multiple Regression
20 questions · exam conditions
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Multiple RegressionQuestion 1 of 20

An economist estimates a multiple regression model: GDP_Growth = β₀ + β₁(Investment_Rate) + β₂(Education_Index) + β₃(Trade_Openness) + ε. The model achieves R² = 0.72 with 45 observations. When a fourth variable (Infrastructure_Quality) is added, R² increases to 0.74. What conclusion about model improvement is most appropriate?

The infrastructure variable significantly improves model performance since R² increased and should be retained permanently
The modest R² increase suggests infrastructure quality provides meaningful but limited additional explanatory power
The improvement should be evaluated using adjusted R² or F-test since R² always increases with additional variables
The 0.02 increase indicates strong practical significance and justifies the more complex four-variable specification
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Business Statistics Quiz

Business Statistics Quiz: Multiple Regression

Practice Multiple Regression in Business Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Multiple Regression, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

An economist estimates a multiple regression model: GDP_Growth = β₀ + β₁(Investment_Rate) + β₂(Education_Index) + β₃(Trade_Openness) + ε. The model achieves R² = 0.72 with 45 observations. When a fourth variable (Infrastructure_Quality) is added, R² increases to 0.74. What conclusion about model improvement is most appropriate?

  1. The infrastructure variable significantly improves model performance since R² increased and should be retained permanently
  2. The modest R² increase suggests infrastructure quality provides meaningful but limited additional explanatory power
  3. The improvement should be evaluated using adjusted R² or F-test since R² always increases with additional variables (correct answer)
  4. The 0.02 increase indicates strong practical significance and justifies the more complex four-variable specification
Explanation: R² always increases (or stays the same) when variables are added, regardless of whether they truly improve the model. A 0.02 increase from 0.72 to 0.74 with one additional variable and 45 observations may not represent genuine improvement. Adjusted R² or an F-test for the added variable would properly account for the loss of degrees of freedom. Option A is wrong because R² increase alone doesn't indicate significance. Option B assumes the increase is meaningful without proper testing. Option D overstates the practical significance of a 0.02 increase.

Question 2

An operations researcher develops a model: Productivity = α + β₁(Experience) + β₂(Training) + β₃(Equipment_Age) + ε, using data from 60 workers. The F-statistic for overall model significance is 12.8. If the critical F-value at 5% significance level is 2.76, and individual t-tests show that only Experience and Training are significant, what conclusion is most appropriate?

  1. Remove Equipment_Age immediately since it lacks individual significance, then re-estimate the model with remaining variables
  2. The model is overall significant and useful for prediction, though Equipment_Age may not contribute meaningfully (correct answer)
  3. Individual insignificance of Equipment_Age contradicts overall model significance, indicating specification errors
  4. Retain all variables since overall F-test significance overrides individual t-test results for model validity
Explanation: The F-statistic of 12.8 exceeds the critical value of 2.76, so the overall model is statistically significant, meaning at least one predictor is related to productivity. However, Equipment_Age being individually non-significant suggests it may not add meaningful explanatory power. This situation is common and doesn't contradict overall significance. Option A is too hasty - variable removal should consider theory and adjusted R². Option C is wrong because individual non-significance doesn't contradict overall significance when other variables are significant. Option D is wrong because overall significance doesn't automatically justify retaining non-significant individual variables.

Question 3

A supply chain manager uses multiple regression to predict delivery time (in days) based on distance (X₁, in miles), package weight (X₂, in pounds), and weather severity index (X₃, scale 1-10). The model equation is: Time = 2.1 + 0.005X₁ + 0.02X₂ + 0.8X₃. If the correlation between distance and weather severity is 0.65, what interpretation concern should the manager prioritize?

  1. Multicollinearity may inflate standard errors and make coefficient estimates unreliable for policy decisions (correct answer)
  2. The high correlation indicates model specification error requiring interaction terms between distance and weather variables
  3. Weather severity should be removed since its correlation with distance creates redundant predictive information
  4. The correlation suggests heteroscedasticity problems that will bias coefficient estimates toward zero
Explanation: A correlation of 0.65 between predictors indicates moderate to strong correlation, which raises multicollinearity concerns. Multicollinearity inflates standard errors, making coefficient estimates less reliable and hypothesis tests less powerful. Option B is wrong because correlation between predictors doesn't necessarily indicate need for interaction terms. Option C is wrong because 0.65 correlation, while concerning, doesn't automatically mean one variable should be removed. Option D is wrong because correlation between predictors relates to multicollinearity, not heteroscedasticity (which concerns error variance patterns).

Question 4

A multiple regression model is built to predict the sale price of a home. The model's Standard Error of the Estimate (also known as Residual Standard Error) is $15,250. Which of the following is the best interpretation of this value?

  1. The average of all the model's prediction errors (residuals) is $15,250.
  2. The standard deviation of the estimated slope coefficients in the model is, on average, $15,250.
  3. 95% of the model's predictions will fall within $15,250 of the actual home prices.
  4. A typical prediction error made by the model is approximately $15,250. (correct answer)
Explanation: The Standard Error of the Estimate (SEE) is the standard deviation of the residuals. It represents the typical size of a prediction error (a residual), measured in the units of the dependent variable. It is a measure of the typical vertical distance between the observed data points and the regression plane. It is not the average error (which is zero by definition) nor is it related to the standard errors of the coefficients. It represents one standard deviation of error, not a 95% interval.

Question 5

A simple linear regression model is fit to a dataset. Later, a single data point is added. This new point has a value for the independent variable that is far from the mean of the other data points, but it lies very close to the originally fitted regression line. Which of the following is the most likely impact of adding this point and refitting the model?

  1. The slope of the regression line will change significantly, but the R-squared value will decrease.
  2. The R-squared value will likely increase, and the standard errors of the coefficients will likely decrease. (correct answer)
  3. The point will be identified as an influential outlier, and the regression coefficients will become biased.
  4. The intercept will change significantly, but the slope will remain exactly the same.
Explanation: This point is a 'good' leverage point. Because its x-value is extreme, it has high leverage. Because it lies close to the existing regression line, it confirms the trend observed in the rest of the data. Adding such a point will anchor the regression line more firmly, reducing the uncertainty in the slope estimate and thus decreasing the standard errors of the coefficients. By extending the range of X and fitting the pattern well, it will also typically increase the R-squared value.

Question 6

A financial analyst develops two regression models to predict a company's stock return, using a sample of 100 observations.

  • Model A: Uses 3 predictors and has an R-squared of 0.650.
  • Model B: Uses the same 3 predictors as Model A plus 2 additional predictors that are only weakly correlated with the stock return.

If Model B has an R-squared of 0.660, which statement is the most accurate comparison of the two models?

  1. Model B is superior because it has a higher R-squared, indicating it explains more variance.
  2. Model A is likely preferable because its adjusted R-squared is probably higher than Model B's. (correct answer)
  3. The models are practically identical, and either can be used since the change in R-squared is small.
  4. Neither model is reliable, as adding predictors should result in a much larger increase in R-squared.
Explanation: Adjusted R-squared is used to compare models with different numbers of predictors. It penalizes the model for adding predictors that do not significantly improve the fit. While R-squared will always increase (or stay the same) when a new predictor is added, adjusted R-squared will only increase if the added predictor improves the model more than would be expected by chance. A small increase in R-squared (from 0.650 to 0.660) from adding two predictors is unlikely to overcome the penalty, meaning Model A's adjusted R-squared is likely higher than Model B's, making it the more parsimonious and preferable model.

Question 7

A researcher models employee salary (Y) using years of experience (X1X_1). The estimated model is Y^=b0+b1X1\hat{Y} = b_0 + b_1 X_1, and b1b_1 is found to be positive and significant. The researcher realizes that "innate ability" (X2X_2), a relevant predictor of salary, was omitted from the model. It is known that innate ability is positively correlated with salary and also positively correlated with years of experience (more able people may stay in their field longer). What is the likely effect of omitting "innate ability" on the estimated coefficient b1b_1?

  1. The coefficient b1b_1 is likely overestimated. (correct answer)
  2. The coefficient b1b_1 is likely underestimated.
  3. The coefficient b1b_1 is unaffected, but its standard error is inflated.
  4. The intercept b0b_0 is biased, but the slope coefficient b1b_1 remains unbiased.
Explanation: This is a case of omitted variable bias. The bias in b1b_1 is determined by the product of two effects: the effect of the omitted variable (ability) on the dependent variable (salary), and the correlation between the omitted variable (ability) and the included variable (experience). Here, ability has a positive effect on salary, and ability is positively correlated with experience. The product of these two positive effects is positive, meaning the estimated coefficient b1b_1 has a positive bias. Therefore, b1b_1 is likely overestimated.

Question 8

An analyst models the daily number of customers at a restaurant. One predictor is "Season," a categorical variable. The model uses "Winter" as the reference category and includes dummy variables for "Spring," "Summer," and "Fall." The fitted model equation is:

Customers^=180−20(Spring)+50(Summer)+10(Fall)+…\widehat{\text{Customers}} = 180 - 20(\text{Spring}) + 50(\text{Summer}) + 10(\text{Fall}) + \dots

How should the coefficient for "Summer" (50) be interpreted?

  1. The model predicts that, holding other factors constant, there are 50 customers on an average day in Summer.
  2. The model predicts that, holding other factors constant, there are 50 more customers on an average day in Summer compared to Winter. (correct answer)
  3. The model predicts that, holding other factors constant, there are 70 more customers on an average day in Summer compared to Spring.
  4. The model predicts that, holding other factors constant, there are 50 more customers on an average day in Summer compared to the average of all other seasons.
Explanation: In a multiple regression with dummy variables, the coefficient for a particular category represents the estimated difference in the mean response between that category and the reference (or baseline) category, holding all other variables constant. In this model, "Winter" is the reference category. Therefore, the coefficient of 50 for "Summer" means that the model predicts an average of 50 more customers per day in Summer than in Winter.

Question 9

A marketing analyst models the natural logarithm of sales (ln(Sales)) as a function of advertising spending (AdSpend, measured in thousands of dollars). The fitted equation is:

ln⁡(Sales)^=5.2+0.04×AdSpend\widehat{\ln(\text{Sales})} = 5.2 + 0.04 \times \text{AdSpend}

How should the coefficient 0.04 be interpreted?

  1. For each additional $1,000 in advertising spending, sales are expected to increase by $0.04.
  2. For each 1% increase in advertising spending, sales are expected to increase by 0.04%.
  3. For each additional $1,000 in advertising spending, sales are expected to increase by approximately 4%. (correct answer)
  4. For each additional $1,000 in advertising spending, the natural log of sales is expected to increase by 4.
Explanation: This is a log-level model. The interpretation for a coefficient β1\beta_1 in such a model is that a one-unit change in the independent variable is associated with a 100×β1100 \times \beta_1% change in the dependent variable. Here, AdSpend is in thousands of dollars, so a one-unit change is an additional $1,000. The coefficient is 0.04. Therefore, an additional $1,000 in advertising spend is associated with an approximate 100×0.04=4100 \times 0.04 = 4% increase in sales.

Question 10

A researcher wants to determine if adding two new variables, X3X_3 and X4X_4, improves a regression model that already contains X1X_1 and X2X_2. The "reduced" model (with X1,X2X_1, X_2) has an R-squared of 0.40, while the "full" model (with X1,X2,X3,X4X_1, X_2, X_3, X_4) has an R-squared of 0.45. A partial F-test is conducted to test the joint significance of X3X_3 and X4X_4 (H0:β3=β4=0H_0: \beta_3 = \beta_4 = 0). The p-value for this F-test is found to be 0.03. At a 5% significance level, what is the correct interpretation?

  1. The null hypothesis is rejected, indicating that the full model provides a statistically significant improvement in fit over the reduced model. (correct answer)
  2. Since the increase in R-squared is only 0.05, the new variables are not practically significant and the reduced model is better.
  3. The null hypothesis is accepted, meaning the two new variables do not contribute significantly to the model.
  4. At least one of the two new variables must be individually significant with a p-value less than 0.05.
Explanation: When you encounter questions about comparing nested regression models, you're dealing with partial F-tests that assess whether adding variables significantly improves model performance beyond what you'd expect by chance. The partial F-test here tests H0:β3=β4=0H_0: \beta_3 = \beta_4 = 0 (the new variables have no effect) against H1H_1: at least one coefficient is non-zero. With a p-value of 0.03, which is less than the 0.05 significance level, you reject the null hypothesis. This means the data provides strong evidence that adding X3X_3 and X4X_4 creates a statistically significant improvement over the reduced model. Answer A correctly captures this interpretation. Answer B makes the common mistake of confusing statistical significance with practical significance. While the R-squared increase of 0.05 might seem small, statistical significance is determined by the F-test p-value, not the magnitude of R-squared change. Answer C gets the hypothesis testing backwards—when p < α, you reject (not accept) the null hypothesis. Answer D confuses joint significance with individual significance. A significant partial F-test only tells you that at least one variable contributes significantly as a group; it doesn't guarantee that each variable would be individually significant in separate t-tests. Remember that partial F-tests evaluate the collective contribution of multiple variables simultaneously. Focus on comparing the p-value to your significance level—if p < α, the additional variables provide statistically significant improvement, regardless of how small the R-squared increase appears.

Question 11

An analyst begins with a properly specified regression model predicting sales. They then add a new predictor variable that is truly random noise and has no theoretical or actual relationship with sales or any other predictors. What is the most likely consequence of adding this irrelevant variable and refitting the model?

  1. The value of R-squared will decrease, and the F-statistic for the overall model will decrease.
  2. The coefficients of the original variables will become biased due to the inclusion of the new variable.
  3. The value of adjusted R-squared will likely decrease, and the standard errors of the other coefficients may increase. (correct answer)
  4. There will be no change to any of the model's statistics since the new variable is completely irrelevant.
Explanation: Adding any variable, even an irrelevant one, will cause the R-squared to increase slightly (or stay the same) due to chance correlation in the sample. However, adjusted R-squared, which penalizes for adding predictors, will likely decrease because the small gain in R-squared is not enough to offset the penalty. Furthermore, adding an irrelevant variable uses up a degree of freedom from the error term, which can lead to a slight increase in the standard errors of the other coefficients, reducing their precision.

Question 12

A regression model is used to predict household electricity consumption based on income. After fitting the model, a plot of the residuals versus the predicted values shows a distinct cone shape, with the spread of the residuals increasing as the predicted consumption increases. What is the primary implication of this finding?

  1. The regression coefficients are biased, consistently overestimating the true population effects.
  2. The standard errors of the regression coefficients are unreliable, making hypothesis tests and confidence intervals invalid. (correct answer)
  3. The assumption of normality of errors has been violated, suggesting the dependent variable should be transformed.
  4. The linear model is inappropriate; a non-linear relationship exists between income and consumption.
Explanation: A cone-shaped pattern in the residual plot (residuals vs. predicted values) is the classic sign of heteroscedasticity, which means the variance of the errors is not constant. While heteroscedasticity does not cause bias in the coefficient estimates themselves, it violates the assumption of constant variance. This leads to incorrect (typically underestimated) standard errors for the coefficients, which in turn makes t-tests, p-values, and confidence intervals unreliable and invalid.

Question 13

A marketing analyst runs a multiple regression to predict monthly sales. The model includes several predictors related to advertising spend. The overall model has a high adjusted R-squared of 0.85 and a statistically significant F-test (p < 0.001). However, none of the individual predictor variables have statistically significant t-statistics (all p > 0.10). Which of the following is the most likely explanation for this outcome?

  1. The overall model is not a good fit for the data.
  2. One or more important predictor variables have been omitted from the model.
  3. High multicollinearity exists among the predictor variables. (correct answer)
  4. The sample size is too small to draw any valid conclusions.
Explanation: This is a classic symptom of severe multicollinearity. When predictors are highly correlated, the model as a whole can have strong explanatory power (high R-squared, significant F-test), but the standard errors of the individual coefficients become inflated. This makes it difficult to disentangle the unique contribution of each predictor, leading to insignificant t-tests. The model can't tell which of the correlated predictors is 'responsible' for the effect.

Question 14

To analyze the relationship between a company's advertising expenditures and its revenue, an economist fits a log-log regression model using data from 50 different companies:

ln⁡(Revenue)^=2.5+1.1×ln⁡(Advertising)\widehat{\ln(\text{Revenue})} = 2.5 + 1.1 \times \ln(\text{Advertising})

How is the coefficient 1.1 correctly interpreted in this context?

  1. A 1% increase in advertising expenditure is associated with an estimated 1.1% increase in revenue. (correct answer)
  2. For every one-dollar increase in advertising, revenue is expected to increase by 1.1%.
  3. A 1% increase in advertising is associated with a 1.1 unit increase in the natural log of revenue.
  4. The elasticity of revenue with respect to advertising is estimated to be 2.5.
Explanation: In a log-log regression model (where both the dependent and independent variables are in natural logarithms), the slope coefficient represents the elasticity. Elasticity is the ratio of the percentage change in the dependent variable for a percentage change in the independent variable. Therefore, a coefficient of 1.1 means that a 1% increase in the independent variable (Advertising) is associated with an estimated 1.1% increase in the dependent variable (Revenue).

Question 15

An analyst is concerned about multicollinearity in a regression model with four predictors: X1,X2,X3,X4X_1, X_2, X_3, X_4. The Variance Inflation Factors (VIFs) for the predictors are calculated as follows: VIF(X1X_1) = 1.2, VIF(X2X_2) = 8.9, VIF(X3X_3) = 15.4, and VIF(X4X_4) = 1.5. Based on these VIFs, what is the most pressing concern for the analyst?

  1. The model's overall predictive power is severely compromised due to the high VIF for X3X_3.
  2. The coefficient for X3X_3 is biased and cannot be trusted as an estimate of the true effect.
  3. The standard error for the coefficient of X3X_3 is significantly inflated, making its individual effect difficult to estimate precisely. (correct answer)
  4. Predictor X3X_3 is not linearly related to the dependent variable and should be removed from the model.
Explanation: The Variance Inflation Factor (VIF) measures how much the variance of an estimated regression coefficient is increased because of multicollinearity. A common rule of thumb is that a VIF greater than 10 (as for X3X_3) indicates a problematic level of multicollinearity. The direct consequence is not bias in the coefficient estimate but a large inflation of its standard error, which widens the confidence interval and makes the estimate unreliable and unstable.

Question 16

A regression model is created to predict the monthly salary of employees based on their years of experience. For an employee with 10 years of experience, the model predicts a salary of 80,000.The9580,000. The 95% confidence interval for the mean salary of *all* employees with 10 years of experience is [78,000, 82,000].The9582,000]. The 95% prediction interval for the salary of a *specific, individual* employee with 10 years of experience is [72,000, $88,000]. Why is the prediction interval significantly wider than the confidence interval?

  1. The prediction interval must account for both the uncertainty in estimating the mean response and the random variation of individual responses. (correct answer)
  2. The confidence interval is narrower because it is based on a larger sample of employees.
  3. The prediction interval is always wider to provide a more conservative estimate and reduce the risk of errors in individual forecasting.
  4. The confidence interval only accounts for sampling error, while the prediction interval also accounts for measurement error in the predictor variable.
Explanation: When you encounter regression problems involving both confidence intervals and prediction intervals, remember that these measure two fundamentally different types of uncertainty around the same predicted value. Both intervals center on the same point estimate ($80,000), but they serve different purposes. A confidence interval estimates the range where the true mean response lies for all individuals with that predictor value. A prediction interval estimates where a single, new observation will fall. The prediction interval is wider because it must account for two sources of uncertainty: the uncertainty in estimating the population mean (which the confidence interval captures) plus the natural variability of individual observations around that mean. Even if we knew the true population mean perfectly, individual employees would still vary randomly around that mean due to factors the model doesn't capture. Looking at the incorrect options: Option B incorrectly suggests sample size differences affect the intervals differently - both use the same data. Option C mischaracterizes the prediction interval as merely being "conservative" rather than accounting for additional variability. Option D incorrectly focuses on measurement error in the predictor variable, which isn't the primary distinction between these intervals. Option A correctly identifies that prediction intervals incorporate both the uncertainty in the mean estimate and the random variation of individual responses around that mean. Study tip: Remember this hierarchy of width: prediction intervals > confidence intervals for mean response > confidence intervals for regression coefficients. Each step adds another layer of uncertainty.

Question 17

A sports analyst is modeling the number of wins for a baseball team. The predictors are Runs Scored (RS) and Runs Allowed (RA). Both variables are known to be highly significant predictors. The analyst then adds a third predictor, Run Differential (RD), which is calculated as RD = RS - RA. The new model is Wins ~ RS + RA + RD. Which of the following is the most likely outcome for the coefficients in this new model?

  1. All three variables will be highly significant with the expected signs (positive for RS and RD, negative for RA).
  2. The R-squared of the model will decrease significantly because the redundant information confuses the model.
  3. The standard errors of the coefficients will become extremely large, and the signs of the coefficients may be counter-intuitive. (correct answer)
  4. The coefficient for RD will be exactly 1, and the coefficients for RS and RA will become zero.
Explanation: This is a case of perfect multicollinearity because one predictor (RD) is a perfect linear combination of two other predictors (RS and RA). In practice, statistical software will either drop one of the variables or fail to run. Conceptually, if the model were to be estimated, the standard errors of the coefficients for RS, RA, and RD would become infinitely large. This makes the coefficient estimates completely unstable and unreliable. It's common in such situations for coefficients to have illogical signs (e.g., wins decreasing with runs scored) because the model cannot disentangle their individual effects.

Question 18

A real estate analyst models house prices using square footage (SQFT, in square feet) and age of the house (AGE, in years). To account for how the market values square footage differently for newer versus older homes, an interaction term is included. The fitted regression equation is:

Price=150,000+120(SQFT)−500(AGE)−2(SQFT×AGE)\text{Price} = 150,000 + 120(\text{SQFT}) - 500(\text{AGE}) - 2(\text{SQFT} \times \text{AGE})

Based on this model, how should the effect of an additional square foot of space be interpreted for a 10-year-old house?

  1. The estimated price increases by $120.
  2. The estimated price increases by $100. (correct answer)
  3. The estimated price increases by $118.
  4. The estimated price decreases by $20.
Explanation: To find the effect of an additional square foot, we need to consider both the main effect of SQFT and its interaction with AGE. The partial derivative of Price with respect to SQFT is 120−2(AGE)120 - 2(\text{AGE}). For a 10-year-old house, AGE = 10. Plugging this into the expression gives the marginal effect of SQFT as 120−2(10)=120−20=100120 - 2(10) = 120 - 20 = 100. Therefore, for a 10-year-old house, each additional square foot is associated with a $100 increase in price.

Question 19

An agronomist models crop yield (in bushels per acre) based on the amount of fertilizer used (F, in pounds per acre). To capture a potential non-linear effect, both F and its square, F2F^2, are included. The fitted model is: Yield^=10+5.2F−0.3F2\widehat{\text{Yield}} = 10 + 5.2F - 0.3F^2 What does this model imply about the relationship between fertilizer and yield?

  1. The relationship is negative; as more fertilizer is added, the yield decreases due to the negative coefficient on the squared term.
  2. For every one-pound increase in fertilizer, the yield is estimated to increase by 4.9 bushels per acre.
  3. The effect of additional fertilizer on yield is initially positive but diminishes and eventually becomes negative as more fertilizer is applied. (correct answer)
  4. The model is misspecified because the coefficients for F and F2F^2 must have the same sign to be interpretable.
Explanation: This is a quadratic model. The positive coefficient on F (5.2) indicates that at low levels of fertilizer, increasing F leads to an increase in yield. The negative coefficient on F2F^2 (-0.3) indicates that the relationship is concave down (an inverted U-shape). This means the positive effect of fertilizer diminishes as more is applied, eventually reaching a peak, after which additional fertilizer will lead to a decrease in yield. This is known as the law of diminishing returns.

Question 20

A multiple regression model with five predictor variables is fitted to a dataset. The overall F-statistic for the model is 25.6 with a p-value of < 0.001. The t-statistics for the five predictors have p-values of 0.02, 0.04, 0.35, 0.41, and 0.58, respectively. What is the most appropriate conclusion based on this information?

  1. The model is invalid because three of the five predictors are not statistically significant.
  2. The high F-statistic indicates that the predictors, as a group, explain a large and practically significant proportion of the variance.
  3. Only the first two predictors are useful and the model should be refit with only those variables.
  4. There is strong evidence that the set of five predictor variables, as a group, is useful for predicting the response. (correct answer)
Explanation: The overall F-test assesses the null hypothesis that all regression coefficients are simultaneously equal to zero (H0:β1=β2=⋯=β5=0H_0: \beta_1 = \beta_2 = \dots = \beta_5 = 0). A small p-value for the F-test (like < 0.001) provides strong evidence to reject this null hypothesis. This means that at least one of the predictor variables is linearly related to the dependent variable. It validates the usefulness of the model as a whole, even if not all individual predictors are significant.