Source: APEC 3002, Applied Microeconomics: Managerial Economics, University of Minnesota Twin Cities
Difficulty: Intermediate | Prerequisites: Basic statistics (mean, variance), introductory econometrics concepts, familiarity with Excel regression tools.
This lab applies multiple linear regression to a real managerial question: does advertising spending drive product sales, and does price matter? It sits at the intersection of statistics and business decision-making, the heart of managerial economics.
You need to be comfortable reading regression output (coefficients, R-squared, standard errors) and conducting t-tests before this material will click. If you can interpret a simple regression, this lab extends that to two explanatory variables and teaches you how to evaluate whether each one pulls its weight statistically.
The broader lesson: knowing that a variable appears in your model is not the same as knowing it matters. The t-test is how you tell the difference.
A multiple regression of Sales on Advertising (A) and Price (P) produced an R-squared of 0.969, meaning the model explains about 97% of the variation in sales. T-tests showed advertising is statistically significant at the 95% level, but price is not. The regression equation is S = 284.81 + 0.5497A − 1.2716P.
Multiple linear regression
A statistical method that models the relationship between one dependent variable and two or more independent variables. In simple terms, it lets you measure how several factors (here, advertising and price) jointly influence an outcome (sales).
R-squared (R²)
The proportion of variation in the dependent variable that is explained by the independent variables, ranging from 0 to 1. Think of it as a score for how well your model fits the data. An R² of 0.969 means the model captures 96.9% of the movement in sales.
Adjusted R-squared
A modified version of R² that penalises the addition of variables that do not improve the model. In simple terms, it stops you from inflating your fit score by throwing in irrelevant variables. Here, adjusted R² = 0.9603.
Degrees of freedom (df)
The number of observations minus the number of estimated parameters. With 10 observations and 3 coefficients (intercept, A, P), degrees of freedom = 10 − 3 = 7. This number feeds into the t-distribution used for hypothesis testing.
t-statistic (t-calc)
The ratio of a coefficient to its standard error. It measures how many standard errors the estimated coefficient is from zero. A larger absolute value means the coefficient is more likely to be statistically different from zero.
Critical t-value (t-crit)
The threshold from the t-distribution at a chosen confidence level (here, 95%) and a given number of degrees of freedom. If the absolute value of t-calc exceeds t-crit, you reject the null hypothesis. For 7 df at 95%, t-crit = 2.365.
Null hypothesis (H₀)
The default assumption that a coefficient equals zero, meaning the variable has no effect on the dependent variable. Think of it as the "nothing is going on" baseline you are trying to disprove.
Alternative hypothesis (H₁)
The claim that the coefficient is not equal to zero, i.e. the variable does have an effect. Rejecting H₀ supports H₁.
P-value
The probability of observing a t-statistic at least as extreme as the one calculated, assuming H₀ is true. A p-value below 0.05 (for a 95% confidence level) means you reject H₀. The p-value for advertising here is 0.0000; for price, 0.1930.
Standard error
A measure of the variability of a coefficient estimate. Smaller standard errors mean the estimate is more precise. Used as the denominator when computing the t-statistic.
Residual
The difference between the observed value and the value predicted by the regression model. Residuals that scatter randomly around zero suggest the model is well-specified.
F-statistic
Tests whether the overall regression model is statistically significant, i.e. whether at least one independent variable has a non-zero coefficient. Here, F = 109.87 with a significance of 0.0000, so the model as a whole is significant.
The dependent variable is Sales (S).
The independent variables are Advertising expenditure (A) and Price (P).
The model takes the form: S = β₀ + β₁A + β₂P + ε, where ε is the error term.
Data: 10 observations.
Intercept (β₀): 284.8099. This is the predicted sales level when both advertising and price are zero. In practice, it anchors the equation rather than carrying a useful standalone interpretation.
Advertising coefficient (β₁): 0.5497. For every additional dollar spent on advertising, sales increase by approximately 0.55 units, holding price constant.
Standard error: 0.0416
t-stat: 13.2243
p-value: 0.0000
Price coefficient (β₂): −1.2716. For every dollar increase in price, sales decrease by about 1.27 units, holding advertising constant. However, this coefficient is not statistically significant (see below).
Standard error: 0.8830
t-stat: −1.4401
p-value: 0.1930
R² = 0.9691. The model explains 96.9% of the variation in sales.
Adjusted R² = 0.9603. After penalising for the number of variables, fit remains strong.
F-statistic = 109.87, Significance F = 0.0000. The model overall is highly significant.
The question each t-test answers: is this coefficient statistically different from zero at the 95% confidence level?
Setup:
H₀: βᵢ = 0 (the variable has no effect)
H₁: βᵢ ≠ 0 (the variable does have an effect)
Critical value: t-crit = 2.365 (two-tailed, 7 df, 95%)
Results:
Intercept: t-calc = 3.623, which exceeds 2.365. Reject H₀. The intercept is significantly different from zero.
Advertising (A): t-calc = 13.224, which far exceeds 2.365. Reject H₀. Advertising has a statistically significant positive effect on sales.
Price (P): t-calc = −1.440, whose absolute value (1.440) is below 2.365. Fail to reject H₀. Price is not statistically significant at the 95% level in this sample.
Failing to reject H₀ for price does not prove that price has no effect on sales. It means the data in this sample do not provide strong enough evidence to conclude that price significantly influences sales once advertising is already in the model. The coefficient is negative (as expected), but the standard error is large relative to the estimate, so the signal is drowned by noise.
The regression output includes a residual table showing Predicted S and Residuals for each of the 10 observations.
Residuals range from roughly −44.05 to +39.54.
Residual plots for both A and P show residuals scattered around zero with no obvious pattern, suggesting the linear model is a reasonable fit and there are no glaring violations of the regression assumptions (e.g. heteroskedasticity or non-linearity).
S = 284.81 + 0.5497A - 1.2716Pt = \frac{\hat{\beta}_i}{SE(\hat{\beta}_i)}Example for advertising:
t_A = \frac{0.5497}{0.0416} = 13.2243\text{Degrees of freedom} = n - k = 10 - 3 = 7Where n = number of observations and k = number of estimated coefficients (intercept + 2 independent variables).
R^2 = 1 - \frac{SS_{\text{Residual}}}{SS_{\text{Total}}} = 1 - \frac{6343.18}{205466.26} = 0.9691Firms routinely use regression models like this to decide how to allocate marketing budgets. If advertising has a strong, statistically significant coefficient, it provides evidence that increasing ad spend will lift sales, which feeds directly into return-on-investment calculations.
The finding that price is statistically insignificant here is also practically useful: it may signal that the product is in an inelastic region of demand (explored further in the companion notes on elasticity), or that the sample is too small or the price range too narrow to detect the effect.
"Fail to reject means the variable has no effect." It does not. It means the evidence is insufficient to conclude an effect at the chosen significance level. With more data or a wider price range, the result could change.
"A high R² means every variable in the model matters." R² measures overall model fit, not the significance of individual variables. Price contributes to R² but still fails its t-test.
"The coefficient size tells you which variable is more important." Coefficients are not directly comparable when variables have different units. A £0.55 increase per advertising dollar vs. a £1.27 decrease per price dollar does not mean price matters more, especially when price is not even significant.
"Rejecting the null hypothesis proves the alternative is true." Rejection means the data are inconsistent with H₀ at a given confidence level. It is not absolute proof.
⚠️ You will almost certainly be asked to interpret regression output: know how to read coefficients, R², and p-values from a summary table.
⚠️ Expect a question on how to conduct and interpret a t-test. Be able to state H₀ and H₁, compare t-calc to t-crit, and draw the correct conclusion.
⚠️ The distinction between "reject" and "fail to reject" is a perennial exam favourite. Never say "accept the null hypothesis."
⚠️ Degrees of freedom (n − k) feeds into looking up the correct t-crit. An error here cascades through the whole test.
⚠️ Know the difference between statistical significance (t-test result) and practical significance (the size of the effect and whether it matters for business decisions).
True or False: An R² of 0.969 means the model perfectly explains all variation in sales.
False. It explains 96.9%, not 100%.
Fill in the blank: The degrees of freedom for this regression are ____.
7 (10 observations minus 3 coefficients).
True or False: If the absolute value of t-calc is less than t-crit, you reject the null hypothesis.
False. You fail to reject.
Fill in the blank: The p-value for the advertising coefficient is ____, indicating it is ____ at the 95% level.
0.0000; statistically significant.
True or False: "Fail to reject H₀" is the same as "accept H₀."
False. You never "accept" the null; you simply lack sufficient evidence to reject it.
Q: Given the regression S = 284.81 + 0.5497A − 1.2716P, what is the predicted sales level when A = 900 and P = 110?
A: S = 284.81 + 0.5497(900) − 1.2716(110) = 284.81 + 494.73 − 139.88 = 639.66 (approximately 639.65, matching the lab's Table 3 row for A = 900, P = 110).
Q: A t-test for the coefficient on price yields a t-calc of −1.440 and the critical value at 95% with 7 df is 2.365. What is your conclusion?
A: The absolute value of t-calc (1.440) is less than t-crit (2.365), so we fail to reject H₀. There is insufficient evidence at the 95% confidence level to conclude that price has a statistically significant effect on sales.
Q: Why is it incorrect to say "we accept the null hypothesis"?
A: Failing to reject H₀ means the data are not strong enough to rule it out; it does not provide evidence that H₀ is true. Future data or a different sample could yield a different result.
Q: The F-statistic for the overall model is 109.87 with a significance of 0.0000. What does this tell you?
A: At least one of the independent variables (A or P) has a non-zero coefficient. The model as a whole has significant explanatory power.
Q: If you added a third independent variable and R² increased slightly but adjusted R² decreased, what would that suggest?
A: The new variable does not improve the model enough to justify its inclusion. The marginal gain in fit is outweighed by the penalty for adding another parameter.
This material connects directly to price elasticity of demand, covered in the companion study notes for Part 4 of this lab. The regression equation estimated here is the same one used to compute elasticity at different price and advertising levels.
It also ties into demand estimation more broadly: the regression is modelling a demand function, and the coefficients reflect the marginal effect of each variable on quantity demanded.
If you go on to study omitted variable bias, this lab is a good case study: dropping advertising from the model would likely bias the price coefficient, because advertising and price may be correlated.
Multiple regression, OLS, ordinary least squares, regression analysis, advertising and sales, managerial economics regression, t-test hypothesis testing, R-squared interpretation, degrees of freedom, critical t-value, p-value significance, fail to reject null hypothesis, residual analysis, F-test, ANOVA regression, APEC 3002, applied microeconomics, demand estimation, regression coefficients, standard error, statistical significance vs practical significance