Business Statistics Quiz: Interaction Terms

20 questions · exam conditions

0:00

Interaction TermsQuestion 1 of 20

A financial analyst models stock returns ( $Y$ ) using market volatility ( $X_1$ ) and firm size ( $X_2$ , logged): $\hat{Y} = 0.08 - 0.3X_1 + 0.02X_2 - 0.05X_1X_2$ . For a firm with log(size) = 4, if market volatility increases from 0.2 to 0.3, what is the predicted change in stock returns?

Returns decrease by 0.12

Returns decrease by 0.10

Returns decrease by 0.08

Returns decrease by 0.05

← Back to quizzes

Business Statistics Quiz

Business Statistics Quiz: Interaction Terms

Practice Interaction Terms 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 Interaction Terms, 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

Returns decrease by 0.12
Returns decrease by 0.10
Returns decrease by 0.08
Returns decrease by 0.05 (correct answer)

Explanation: The marginal effect of volatility for a firm with log(size) = 4 is ∂Y/∂X₁ = -0.3 - 0.05(4) = -0.3 - 0.2 = -0.5. When volatility increases by 0.1 (from 0.2 to 0.3), the predicted change is -0.5 × 0.1 = -0.05. Choice A incorrectly uses -0.3 - 0.05 = -0.35 then multiplies by 0.1 and adds extra terms. Choice B uses only the main effect coefficient. Choice C makes an arithmetic error in calculating the interaction effect.

Question 2

A retail analyst estimates the model $\text{Sales} = \beta_0 + \beta_1 \text{Price} + \beta_2 \text{Quality} + \beta_3 \text{Price} \times \text{Quality} + \varepsilon$ . The estimated coefficients are $\hat{\beta_1} = -2.0$ , $\hat{\beta_2} = 1.5$ , and $\hat{\beta_3} = 0.3$ . At what quality level does price have zero effect on sales?

Quality = 6.67 (correct answer)
Quality = 5.00
Quality = 2.00
Quality = 1.50

Explanation: The marginal effect of price on sales is ∂Sales/∂Price = -2.0 + 0.3(Quality). Setting this equal to zero: -2.0 + 0.3(Quality) = 0, which gives Quality = 2.0/0.3 = 6.67. Choice B incorrectly uses 1.5/0.3. Choice C uses 2.0 without dividing by 0.3. Choice D simply uses the β₂ coefficient value.

Question 3

An HR department models employee salary (in thousands of dollars) based on years of experience (Experience) and whether the employee has an advanced degree (AdvancedDegree, a dummy variable equal to 1 for yes, 0 for no). The fitted model is:

$\widehat{\text{Salary}} = 45 + 2.1(\text{Experience}) + 10.5(\text{AdvancedDegree}) + 0.8(\text{Experience} \times \text{AdvancedDegree})$

According to this model, what is the estimated additional annual salary an employee with an advanced degree earns for each additional year of experience?

$2,100
$2,900 (correct answer)
$800
$13,400

Explanation: The effect of an additional year of experience on salary is given by the marginal effect:

\frac{\partial \text{Salary}}{\partial \text{Experience}} = 2.1 + 0.8(\text{AdvancedDegree})

. For an employee with an advanced degree, AdvancedDegree = 1. So, the effect is

2.1 + 0.8(1) = 2.9

. Since salary is in thousands of dollars, this corresponds to $2,900.

Question 4

A marketing analytics firm models monthly sales for a client using the following regression equation, where Sales are in thousands of dollars, AdSpend is in thousands of dollars, and CompIndex is a measure of competitor activity from 1 to 10.

$\text{Sales} = 120 + 4.5(\text{AdSpend}) + 2.0(\text{CompIndex}) - 0.3(\text{AdSpend} \times \text{CompIndex})$

How should the coefficient for AdSpend, $\beta_1 = 4.5$ , be interpreted in the context of this model?

For every $1,000 increase in advertising spending, sales are expected to increase by $4,500, holding the interaction constant.
When there is no competitor activity (i.e., CompIndex = 0), every $1,000 increase in advertising spending is associated with a $4,500 increase in sales. (correct answer)
The average effect of a $1,000 increase in advertising spending is a $4,500 increase in sales, averaged across all levels of competitor activity.
When advertising spending is zero, a one-point increase in the competitor activity index is associated with a $4,500 increase in sales.

Explanation: In a model with an interaction term, the coefficient of a main effect (like

\beta_1

for AdSpend) represents the effect of that variable when the other variable in the interaction term is equal to zero. Here, the effect of AdSpend on Sales is given by

4.5 - 0.3(\text{CompIndex})

. The coefficient

4.5

is the value of this effect when CompIndex = 0.

Question 5

A real estate analyst uses the following model to predict house prices, where Price is in thousands of dollars, SqFt is square footage, and Age is the age of the house in years.

$\widehat{\text{Price}} = 150 + 0.12(\text{SqFt}) - 2.5(\text{Age}) + 0.002(\text{SqFt} \times \text{Age})$

For a 20-year-old house, what is the estimated increase in price for an additional 100 square feet of space?

$12,000
$4,000
$16,000 (correct answer)
$20,000

Explanation: First, find the marginal effect of SqFt on Price, which is

\frac{\partial \text{Price}}{\partial \text{SqFt}} = 0.12 + 0.002(\text{Age})

. Second, substitute Age = 20 into this equation:

0.12 + 0.002(20) = 0.12 + 0.04 = 0.16

. This means for a 20-year-old house, each additional square foot is associated with a $0.16 thousand (or $160) increase in price. Third, calculate the effect for 100 additional square feet:

100 \times \

160 = $16,000$.

Question 6

A researcher models employee productivity (Y) as a function of hours of training (X1) and employee engagement score (X2). The model includes an interaction term. The regression output shows that the p-value for the coefficient of X1 is 0.35, while the p-value for the interaction term X1 × X2 is 0.02.

What is the most valid conclusion regarding the effect of training hours (X1) on productivity?

Training hours do not have a statistically significant effect on productivity because its main effect p-value is greater than 0.05.
The model is misspecified; it should be re-run without the main effect for training hours to follow the principle of parsimony.
The effect of training hours on productivity is statistically significant, but only for certain levels of employee engagement. (correct answer)
The analysis is inconclusive because the p-values for the main effect and the interaction effect contradict each other.

Explanation: When an interaction term is significant, the main effects of the constituent variables cannot be interpreted in isolation. The effect of X1 is

\beta_1 + \beta_3 X_2

. Even if

\beta_1

is not significantly different from zero (p=0.35), the overall effect of X1 can be significant for values of X2 where the combined term

\beta_1 + \beta_3 X_2

is significantly different from zero. The significant interaction (p=0.02) is the key finding.

Question 7

An analyst includes an interaction term (X1 × X2) in a multiple regression model. A diagnostic check reveals that the Variance Inflation Factor (VIF) for the interaction term is 35, while the VIFs for the main effects X1 and X2 are 3 and 4, respectively.

What is the most appropriate action for the analyst to take in response to the high VIF for the interaction term?

Remove the interaction term from the model, as it is causing problematic multicollinearity.
Remove either X1 or X2 from the model to eliminate the source of the high VIF.
Disregard the high VIF for the interaction term, as it is expected and does not invalidate the model's coefficient estimates for the interaction. (correct answer)
Transform all predictor variables using a natural logarithm to reduce the scale and thus the VIF values.

Explanation: It is common and expected for an interaction term to be highly correlated with its constituent main effects, leading to a high VIF. This is often referred to as non-essential multicollinearity. As long as the VIFs for the main effects themselves are not excessively high, this situation is generally not considered problematic for the interpretation of the highest-order term (the interaction). The primary concern of multicollinearity is inflated standard errors, but significance tests on the interaction term are still considered valid. Mean-centering the variables before creating the interaction is a common technique to reduce this type of VIF, but disregarding it is also a valid viewpoint, especially when interpreting the interaction itself.

Question 8

An agronomist studies the effect of a new fertilizer (Fertilizer, kg/hectare) and irrigation level (Irrigation, cm/week) on wheat yield (Yield, tons/hectare). The analysis reveals a statistically significant, negative interaction effect between Fertilizer and Irrigation. Both main effects are positive and significant.

What is the most plausible practical implication of this finding?

The benefits of additional fertilizer are diminished at higher levels of irrigation. (correct answer)
Fertilizer and irrigation should not be used at the same time as they cancel each other out.
Higher irrigation levels consistently increase the positive effect of fertilizer on crop yield.
The model is likely flawed, as fertilizer and water should always have a positive synergistic effect.

Explanation: A negative interaction effect means that the effect of one variable decreases as the other variable increases. Since both main effects are positive, this implies that while both fertilizer and irrigation are individually beneficial, their combined effect is less than the sum of their individual effects. In this context, it suggests that at high levels of irrigation, adding more fertilizer provides less of a yield boost than it does at low levels of irrigation, possibly due to nutrient runoff. This is a classic example of diminishing returns in a multivariate context.

Question 9

An analyst fits two models to predict employee PerformanceScore.

Model A (Main Effects): PerformanceScore ~ YearsExperience + ManagerRating
Model B (Interaction): PerformanceScore ~ YearsExperience + ManagerRating + YearsExperience × ManagerRating

A partial F-test comparing Model A to Model B yields a p-value of 0.015. The adjusted R-squared is 0.54 for Model A and 0.57 for Model B.

Based on these results, which statement represents the most statistically sound decision?

Choose Model A because it is the more parsimonious model and the increase in adjusted R-squared is minimal.
Choose Model B because the interaction term provides a statistically significant improvement in model fit. (correct answer)
Neither model is adequate, as an adjusted R-squared of 0.57 indicates a poor overall fit to the data.
Choose Model A because the interaction in Model B introduces multicollinearity, which invalidates the results.

Explanation: The partial F-test directly compares the explanatory power of the nested models. A p-value of 0.015 is statistically significant at conventional levels (e.g.,

\alpha = 0.05

). This indicates that adding the interaction term (moving from Model A to Model B) results in a significant improvement in the model's ability to explain the variance in PerformanceScore. Therefore, Model B is statistically preferred despite being less parsimonious.

Question 10

A model for predicting monthly sales (in $1000s) for a new product is given by:

$\widehat{\text{Sales}} = 20 + 0.8(\text{OnlineAds}) + 15(\text{InStorePromo}) + 0.3(\text{OnlineAds} \times \text{InStorePromo})$

Where OnlineAds is the online ad budget in $1000s, and InStorePromo is a dummy variable (1 if a promotion is active, 0 otherwise).

In a month with an active in-store promotion and an online ad budget of $30,000, what is the estimated increase in sales compared to a month with the same ad budget but no in-store promotion?

$15,000
$9,000
$24,000 (correct answer)
$35,000

Explanation: The difference in sales between a promotion month and a non-promotion month is found by comparing the predicted sales. Non-promo month:

20 + 0.8(\text{OnlineAds})

. Promo month:

20 + 0.8(\text{OnlineAds}) + 15 + 0.3(\text{OnlineAds}) = 35 + 1.1(\text{OnlineAds})

. The difference is

(35 + 1.1(\text{OnlineAds})) - (20 + 0.8(\text{OnlineAds})) = 15 + 0.3(\text{OnlineAds})

. Plugging in OnlineAds = 30 (since the unit is $1000s), the difference is

15 + 0.3(30) = 15 + 9 = 24

. Since sales are in $1000s, the difference is $24,000.

Question 11

An operations analyst theorizes that the relationship between machine speed (Speed) and output quality (Quality) is quadratic (an inverted U-shape). Furthermore, she suspects this quadratic relationship is fundamentally different for the company's new machines (NewMachine=1) versus its old machines (NewMachine=0).

Which model specification would allow the analyst to test if the entire quadratic relationship between speed and quality differs by machine type?

$\text{Quality} = \beta_0 + \beta_1 \text{Speed} + \beta_2 \text{Speed}^2 + \beta_3 \text{NewMachine} + \epsilon$
$\text{Quality} = \beta_0 + \beta_1 \text{Speed} + \beta_2 \text{NewMachine} + \beta_3 (\text{Speed} \times \text{NewMachine}) + \epsilon$
$\text{Quality} = \beta_0 + \beta_1 \text{Speed} + \beta_2 \text{Speed}^2 + \beta_3 \text{NewMachine} + \beta_4 (\text{Speed} \times \text{NewMachine}) + \epsilon$
$\text{Quality} = \beta_0 + \beta_1 \text{Speed} + \beta_2 \text{Speed}^2 + \beta_3 \text{NewMachine} + \beta_4 (\text{Speed} \times \text{NewMachine}) + \beta_5 (\text{Speed}^2 \times \text{NewMachine}) + \epsilon$ (correct answer)

Explanation: To allow the entire quadratic relationship to differ, the model must allow both the linear and the quadratic terms of Speed to have different coefficients for each machine type. This requires interacting the dummy variable NewMachine with both Speed (the

\beta_4

term) and Speed

^2

(the

\beta_5

term). This model allows for two completely different parabolas for the old and new machines.

Question 12

A consultant models customer churn rate (ChurnRate) based on customer tenure (Tenure, in months) and the number of support tickets filed (Tickets). The estimated model is:

$\widehat{\text{ChurnRate}} = 0.02 - 0.001(\text{Tenure}) + 0.01(\text{Tickets}) + 0.0005(\text{Tenure} \times \text{Tickets})$

At what number of support tickets does an additional month of tenure cease to have a churn-reducing effect (i.e., the effect becomes zero or positive)?

2 tickets (correct answer)
5 tickets
10 tickets
20 tickets

Explanation: First, find the marginal effect of Tenure on ChurnRate:

\frac{\partial \text{ChurnRate}}{\partial \text{Tenure}} = -0.001 + 0.0005(\text{Tickets})

. The churn-reducing effect ceases when this marginal effect is no longer negative (i.e., is

\geq 0

). Set the expression to zero to find the turning point:

-0.001 + 0.0005(\text{Tickets}) = 0

. Solving for Tickets:

0.0005(\text{Tickets}) = 0.001

, which gives Tickets = 2.

Question 13

An HR manager models salary ( $Y$ , in thousands) using education level ( $X_1$ , years) and job performance rating ( $X_2$ , scale 1-10): $\hat{Y} = 30 + 2.5X_1 + 4X_2 + 0.8X_1X_2$ . Two employees differ by 2 years of education and 1 point in performance rating (Employee B has more of both). What is the predicted salary difference between them?

$9,000 higher for Employee B
$11,600 higher for Employee B
$10,600 higher for Employee B (correct answer)
$7,400 higher for Employee B

Explanation: When you encounter a regression model with an interaction term (like

X_1X_2

), you can't simply multiply individual coefficients by the differences. The interaction term means the effect of one variable depends on the level of the other variable. To find the salary difference, calculate the predicted salaries for both employees. Let's say Employee A has

X_1

years of education and performance rating

X_2

. Employee B has

(X_1 + 2)

years and

(X_2 + 1)

performance rating. Employee A:

\hat{Y}_A = 30 + 2.5X_1 + 4X_2 + 0.8X_1X_2

Employee B:

\hat{Y}_B = 30 + 2.5(X_1 + 2) + 4(X_2 + 1) + 0.8(X_1 + 2)(X_2 + 1)

Expanding Employee B's equation:

\hat{Y}_B = 30 + 2.5X_1 + 5 + 4X_2 + 4 + 0.8(X_1X_2 + X_1 + 2X_2 + 2)

\hat{Y}_B = 30 + 2.5X_1 + 4X_2 + 0.8X_1X_2 + 9 + 0.8X_1 + 1.6X_2 + 1.6

The difference is:

\hat{Y}_B - \hat{Y}_A = 9 + 0.8X_1 + 1.6X_2 + 1.6 = 10.6 + 0.8X_1 + 1.6X_2

Wait—this approach gets complex. Instead, the difference is always:

2.5(2) + 4(1) + 0.8(2 \times 1) = 5 + 4 + 1.6 = 10.6

thousand dollars. Answer C (

10,600) is correct. Answer A (

9,000) ignores the interaction term. Answer B (

11,600) likely adds an extra term incorrectly. Answer D (

7,400) probably miscalculates the interaction effect. Remember: with interaction terms, calculate the total effect by considering how both variables change simultaneously, not just their individual effects.

Question 14

A manufacturer models the number of defects per 1000 units (Defects) based on production line speed (Speed, in units per minute) and ambient temperature (Temp, in Celsius). The resulting equation is:

$\widehat{\text{Defects}} = 2.1 + 0.05(\text{Speed}) - 0.1(\text{Temp}) + 0.003(\text{Speed} \times \text{Temp})$

According to the model, how does the relationship between production line speed and the number of defects change as the ambient temperature rises?

As temperature rises, the positive relationship between speed and defects becomes even stronger. (correct answer)
As temperature rises, the positive relationship between speed and defects weakens, and may even become negative.
The relationship between speed and defects is constant and is not influenced by changes in temperature.
As temperature rises, the impact of speed on defects shifts from negative to positive.

Explanation: The marginal effect of Speed on Defects is given by the derivative with respect to Speed:

0.05 + 0.003(\text{Temp})

. Since the coefficient of the interaction term (0.003) is positive, as Temp increases, the entire value of this expression increases. This means that the slope of the relationship between Speed and Defects becomes more positive (stronger) at higher temperatures.

Question 15

A researcher fits a model for income where the predictors Age and YearsOfEducation have been mean-centered. The resulting model is:

$\widehat{\text{Income}} = 65000 + 800(\text{Age_c}) + 2500(\text{Education_c}) + 50(\text{Age_c} \times \text{Education_c})$

Where Age_c = (Age - 42) and Education_c = (Education - 14).

How should the coefficient $\gamma_1 = 800$ for Age_c be interpreted?

For a person with 14 years of education, each additional year of age is associated with an $800 increase in income. (correct answer)
For a person with 0 years of education, each additional year of age is associated with an $800 increase in income.
The average effect of an additional year of age on income, across all education levels, is an increase of $800.
For a 42-year-old person, each additional year of education is associated with an $800 increase in income.

Explanation: When variables are centered, the main effect coefficient for one variable is interpreted as its effect when the other centered variable is equal to zero. Education_c is zero when Education is at its mean, which is 14 years. Therefore,

\gamma_1 = 800

is the effect of a one-year increase in age on income for someone with the mean level of education (14 years).

Question 16

A model is developed to predict sales (Sales) from advertising expenditure (Advertising) and market size (MarketSize). The interaction term Advertising × MarketSize has a p-value of 0.52.

What is the most precise statistical interpretation of this p-value?

It is proven that the effect of advertising on sales is the same regardless of market size.
There is a 52% probability that the interaction effect is zero.
The model provides insufficient evidence to conclude that the effect of advertising on sales varies with market size. (correct answer)
The model should be refit without the interaction term to improve the adjusted R-squared.

Explanation: A high p-value (typically > 0.05) means we fail to reject the null hypothesis. In this context, the null hypothesis is that the interaction coefficient is zero. Failing to reject this null means we do not have enough statistical evidence to say that an interaction exists. It does not prove the null hypothesis is true (Choice A is too strong). Choice B is a common misinterpretation of p-values. Choice D is a possible modeling decision, but not a direct statistical interpretation of the p-value itself.

Question 17

An analyst models customer satisfaction (Satisfaction, scale 1-100) based on product price (Price, in dollars) and whether the customer received premium support (PremiumSupport, a dummy variable).

The analyst generates a plot of the regression lines for both customer groups (premium and standard support). The plot shows two distinct, non-parallel lines, with the line for premium support customers having a less steep negative slope than the line for standard support customers. What does this visualization suggest?

An interaction term between Price and PremiumSupport is likely necessary in the model. (correct answer)
There is significant multicollinearity between Price and PremiumSupport in the model.
The model should only include main effects, as the relationship between price and satisfaction is linear.
Premium support has a constant positive effect on satisfaction, regardless of the product's price.

Explanation: When the relationship between a predictor variable (Price) and the outcome (Satisfaction) differs across levels of another predictor (PremiumSupport), this indicates an interaction effect. Graphically, this is represented by non-parallel regression lines. The fact that the slopes are different means the effect of price depends on the type of support, which is the definition of an interaction.

Question 18

A researcher is modeling Wage based on Education (in years) and Experience (in years). Both predictors are continuous and positively skewed. The researcher is primarily interested in whether the effect of an additional year of experience on wage depends on a person's level of education. There is also concern about potential multicollinearity between the main effects and the interaction term.

What is a common and appropriate first step to prepare the variables for this regression model?

Remove the Education main effect to isolate the impact of the interaction term with Experience.
Mean-center both Education and Experience before creating the interaction term. (correct answer)
Standardize Education but leave Experience in its original units to maintain interpretability.
Combine Education and Experience into a single index variable to simplify the model.

Explanation: Mean-centering (subtracting the mean from each value) the constituent variables before creating their product term is a standard technique to reduce the multicollinearity between the main effects and the interaction term. This does not change the coefficient or significance of the interaction term itself, but it can make the coefficients of the main effects more stable and interpretable.

Question 19

A sociologist models annual income (Income, in $1000s) using dummy variables for gender (IsFemale, 1 if female, 0 if male) and location (IsUrban, 1 if urban, 0 if rural). The fitted model is:

$\widehat{\text{Income}} = 55 - 8(\text{IsFemale}) + 12(\text{IsUrban}) - 4(\text{IsFemale} \times \text{IsUrban})$

Based on this model, what is the correct interpretation of the interaction coefficient, -4?

Urban females earn $4,000 less than rural males, on average.
The income gap between urban and rural residents is $4,000 smaller for females than it is for males.
The income gap between females and males is $4,000 larger in urban areas than it is in rural areas. (correct answer)
Being both female and living in an urban area reduces expected income by $4,000.

Explanation: The interaction coefficient

\beta_3 = -4

represents how the effect of one variable changes based on the other. Let's calculate the gender gap in each location. Rural gender gap (IsUrban=0): (Rural Female) - (Rural Male) =

(\beta_0 + \beta_1) - \beta_0 = \beta_1 = -8

. Urban gender gap (IsUrban=1): (Urban Female) - (Urban Male) =

(\beta_0 + \beta_1 + \beta_2 + \beta_3) - (\beta_0 + \beta_2) = \beta_1 + \beta_3 = -8 - 4 = -12

. The urban gender gap (-$12k) is

4k larger (more negative) than the rural gender gap (-

8k).

Question 20

A financial analyst hypothesizes that a company's stock price becomes more sensitive to news about earnings per share (EPS) during times of high market uncertainty.

To model the stock price, what is the most direct way to incorporate this hypothesis into a regression framework where StockPrice is the dependent variable, EPS_News measures the earnings surprise, and UncertaintyIndex is a measure of market uncertainty?

Include EPS_News and UncertaintyIndex as independent main effect predictors.
Include an interaction term between EPS_News and UncertaintyIndex. (correct answer)
Use UncertaintyIndex as the dependent variable and EPS_News as the predictor.
Create two separate regression models, one for high uncertainty and one for low uncertainty.

Explanation: The hypothesis is that the 'sensitivity' (i.e., the slope or coefficient) of StockPrice with respect to EPS_News changes depending on the level of UncertaintyIndex. This is precisely the definition of an interaction effect. Including an interaction term (EPS_News × UncertaintyIndex) directly models this changing sensitivity within a single equation.