Difficulty: Intermediate | Prerequisites: Basic statistics (mean, variance), simple linear regression, hypothesis testing fundamentals.
This lab sits at the intersection of regression analysis and demand theory. You are using multiple regression to estimate how advertising expenditure and price jointly determine sales, then testing whether each variable matters statistically, and finally computing price elasticities under different scenarios. If you are comfortable with single-variable regression and basic t-tests, this extends those tools to a multi-variable setting with direct managerial applications. The core question: when a firm spends more on advertising, does it sell more, and how sensitive are sales to price changes at different advertising levels?
A multiple regression on 10 observations finds that advertising has a strong, statistically significant positive effect on sales (t = 13.22, well above the critical value of 2.365). Price, surprisingly, fails the t-test at 95% confidence, meaning its coefficient is not statistically distinguishable from zero in this dataset. The model explains about 97% of variation in sales (R² = 0.969). Price elasticity is inelastic (|E| < 1) across all tested scenarios, and higher advertising spending makes demand even less price-sensitive.
Multiple regression
A statistical technique that estimates the relationship between one dependent variable and two or more independent variables simultaneously.
In simple terms, this means you are asking: how do advertising and price together predict sales, rather than looking at each one alone?
R-squared (R²)
The proportion of variance in the dependent variable that is explained by the independent variables in the model. Ranges from 0 to 1.
Think of it as a score for how well the regression line fits the data. An R² of 0.969 means the model accounts for about 97% of the variation in sales.
Degrees of freedom (df)
The number of observations minus the number of estimated parameters (coefficients). With 10 observations and 3 coefficients (intercept, advertising, price), df = 7.
In simple terms, this is how much "room" you have left in your data after fitting the model. More degrees of freedom give your statistical tests more power.
T-test (for regression coefficients)
A hypothesis test that asks whether a given coefficient is statistically different from zero. Compares the calculated t-value to a critical t-value at a chosen confidence level.
Think of it as asking: is this variable's effect real, or could it just be noise?
Critical t-value
The threshold value from the t-distribution. If your calculated t-value exceeds this (in absolute terms), you reject the null hypothesis.
In simple terms, it is the bar your test statistic needs to clear to count as significant.
Null hypothesis (H₀)
The default assumption that a coefficient equals zero, meaning the variable has no effect on the dependent variable.
Price elasticity of demand
A measure of how sensitive quantity demanded is to a change in price. Calculated as the percentage change in quantity demanded divided by the percentage change in price.
Think of it as: if the price goes up by 1%, how much does demand move? An elasticity between 0 and -1 (in absolute value less than 1) means demand is inelastic.
Inelastic demand
Demand where the absolute value of price elasticity is less than 1. Quantity demanded does not change much when price changes.
In simple terms, buyers are not very sensitive to price, so the seller can raise prices without losing much volume.
The regression model estimates Sales as a function of two independent variables: Advertising (A) and Price (P)
The model uses 10 observations to estimate 3 coefficients: the intercept, the advertising coefficient, and the price coefficient
R² = 0.969, meaning the model explains about 97% of the variation in sales data
This is a strong fit, close to 1.0
Only about 3% of variation in sales is unexplained by the model
Degrees of freedom = 7 (10 observations minus 3 estimated coefficients)
Degrees of freedom matter because they determine which t-distribution table you use for hypothesis testing
The scatter plot confirms a visible positive relationship between advertising and sales
The coefficient on price is larger in magnitude than the advertising coefficient, suggesting price has a bigger per-unit effect on sales, though the t-test later reveals the price effect is not statistically significant
The purpose: determine whether each independent variable has a statistically significant effect on sales
Null hypothesis (H₀): the coefficient = 0 (no relationship between the variable and sales)
Alternative hypothesis (H₁): the coefficient ≠ 0 (there is a relationship)
Critical t-value at 95% confidence with 7 df: 2.365
Results from the t-test:
Intercept: calculated t = 3.623 > 2.365, so reject H₀. The intercept is significantly different from zero.
Advertising (A): calculated t = 13.224 > 2.365, so reject H₀. Advertising has a statistically significant positive effect on sales. This is the strongest result in the model.
Price (P): calculated t = -1.440, and |−1.440| < 2.365, so fail to reject H₀. Price does not have a statistically significant effect on sales in this dataset.
The practical interpretation: at the 95% confidence level, we can say advertising drives sales, but we cannot say the same for price. This does not mean price has zero effect, only that the data cannot distinguish its effect from zero with enough confidence.
The lab plots demand curves (Price vs. Quantity Demanded) at three different advertising levels: $600, $900, and $1,200
Each curve is downward-sloping, consistent with the law of demand
Higher advertising shifts the demand curve to the right:
At $600 advertising, demand is lowest across all price points
At $900 advertising, demand shifts rightward
At $1,200 advertising, demand is highest, meaning consumers buy more at every price level
The curves are roughly parallel, suggesting advertising adds a relatively constant boost to quantity demanded regardless of price level
This visual reinforces the regression finding: advertising is the dominant driver of sales in this model
Price elasticity was calculated for five scenarios varying price and advertising levels
All elasticities are between 0 and -1 in absolute value, confirming inelastic demand across the board
Key patterns from the data:
Holding advertising constant at $900, raising price from $90 to $130:
At $90: elasticity = -0.172
At $110: elasticity = -0.219
At $130: elasticity = -0.269
Demand becomes slightly more elastic (more price-sensitive) as price rises, but remains inelastic
Holding price constant at $110, raising advertising from $600 to $1,200:
At $600 advertising: elasticity = -0.295
At $1,200 advertising: elasticity = -0.174
Higher advertising makes demand less elastic (less price-sensitive)
Intuition: heavy advertising builds brand loyalty or perceived value, so consumers become less responsive to price changes
The managerial takeaway: a firm that advertises heavily can raise prices with less risk of losing customers
Multiple regression model:
Sales = β₀ + β₁(Advertising) + β₂(Price) + ε
Where β₀ is the intercept, β₁ and β₂ are the slope coefficients, and ε is the error term.
Degrees of freedom:
df = n - k
Where n = number of observations (10) and k = number of estimated coefficients (3). So df = 7.
T-statistic for each coefficient:
t = (estimated coefficient) / (standard error of the coefficient)
Compare |t| to the critical t-value from the t-distribution table at your chosen confidence level and degrees of freedom.
Decision rule:
If |t-calculated| > t-critical, reject H₀ (the variable is significant). If |t-calculated| ≤ t-critical, fail to reject H₀.
Price elasticity of demand:
Eₚ = (∂Q/∂P) × (P/Q)
In a linear model, ∂Q/∂P is the coefficient on price (β₂). You then multiply by the ratio of price to quantity at the point you are evaluating.
This type of analysis is exactly what marketing departments use to decide advertising budgets. If the regression shows advertising significantly boosts sales but price does not significantly reduce them, a firm might increase ad spend while cautiously raising prices, knowing that demand will hold.
Consumer goods companies (soft drinks, snacks, cosmetics) routinely run these regressions on scanner data to estimate how much a TV or digital ad campaign moved the needle on unit sales, and whether a price promotion was worth the margin hit.
"Fail to reject" means the variable has no effect. It does not. It means the data cannot distinguish the effect from zero at the chosen confidence level. Price may well affect sales, but with only 10 observations the test lacks the power to confirm it.
A high R² means every variable in the model is important. Not necessarily. R² measures overall model fit. Individual variables can be insignificant even in a model with a high R². The price coefficient here is a clear example.
The larger coefficient is always the more important variable. Coefficients depend on the units of measurement. A variable measured in thousands will naturally have a smaller coefficient than one measured in single units. You need to look at the t-statistic (or standardised coefficients) to judge relative importance.
Inelastic demand means price does not matter at all. It means demand is relatively insensitive to price, not completely insensitive. The elasticity values here are small but not zero.
⚠️ Know how to calculate degrees of freedom: n minus the number of estimated coefficients (including the intercept).
⚠️ Be able to compare a calculated t-value against a critical t-value and state the conclusion in proper language ("reject H₀" or "fail to reject H₀," never "accept H₀").
⚠️ Understand what R² tells you and what it does not. It measures fit, not causation, and it does not tell you which individual variables matter.
⚠️ Be ready to interpret price elasticity: know the difference between elastic and inelastic, and be able to explain how advertising shifts the elasticity value.
⚠️ Expect a question that gives you a t-statistic and a critical value and asks you to make the reject/fail-to-reject call. This is a near-certain exam topic.
True or False: An R² of 0.969 means that every variable in the regression is statistically significant.
False. R² measures overall model fit. Individual significance requires a t-test.
Fill in the blank: With 10 observations and 3 coefficients, the degrees of freedom equal ______.
7
True or False: "Fail to reject H₀" is the same as saying the variable has no effect whatsoever.
False. It means the data cannot confirm an effect at the chosen confidence level.
True or False: If price elasticity is -0.22, demand is elastic.
False. |−0.22| < 1, so demand is inelastic.
Fill in the blank: In this lab, increasing advertising from $600 to $1,200 made demand ______ (more/less) price-sensitive.
Less price-sensitive (more inelastic).
Q: A regression of Sales on Advertising (A) and Price (P) yields R² = 0.969. Interpret this value.
A: The model explains approximately 96.9% of the variation in sales. The combination of advertising and price accounts for nearly all observed differences in sales across observations, leaving only about 3.1% unexplained.
Q: The calculated t-value for Advertising is 13.22 and the critical t-value at 95% confidence with 7 degrees of freedom is 2.365. What is the conclusion?
A: Since |13.22| > 2.365, we reject the null hypothesis that the advertising coefficient is zero. Advertising has a statistically significant effect on sales at the 95% confidence level.
Q: The calculated t-value for Price is -1.44. Using the same critical value of 2.365, what is the conclusion, and what does it mean practically?
A: Since |-1.44| = 1.44 < 2.365, we fail to reject the null hypothesis. We cannot conclude that price has a statistically significant effect on sales at the 95% confidence level. This may reflect limited data (only 10 observations) rather than a true absence of price effects.
Q: At a price of $110 and advertising of $900, the price elasticity is -0.219. Is this elastic or inelastic, and what does it imply for the firm?
A: Inelastic, because the absolute value (0.219) is less than 1. A 1% increase in price would reduce quantity demanded by only about 0.22%. The firm could raise prices modestly without a large drop in sales volume.
Q: Explain why increasing advertising from $600 to $1,200 (holding price at $110) changes the price elasticity from -0.295 to -0.174.
A: Higher advertising increases total sales volume (from $474.74 to $804.55). Since the price coefficient (∂Q/∂P) remains constant in a linear model, a larger Q in the denominator of the elasticity formula Eₚ = (∂Q/∂P) × (P/Q) reduces the absolute value of elasticity. Intuitively, advertising builds demand that is less sensitive to price changes.
This connects to demand estimation more broadly: regression is the standard tool for estimating demand functions from data, and everything here (choosing variables, testing significance, computing elasticities) recurs in more complex models with additional variables.
The price elasticity analysis links directly to pricing strategy and revenue maximisation. A firm with inelastic demand can increase total revenue by raising price. Understanding how advertising shifts elasticity ties into optimal advertising budgets, a topic in both managerial economics and marketing analytics.
The t-test framework connects to hypothesis testing throughout statistics and econometrics. The logic (state H₀, compute a test statistic, compare to a critical value, draw a conclusion) is identical whether you are testing regression coefficients, differences in means, or other parameters.
Multiple regression, OLS, ordinary least squares, R-squared, coefficient of determination, goodness of fit, t-test, t-statistic, critical value, degrees of freedom, hypothesis testing, null hypothesis, statistical significance, fail to reject, advertising elasticity, price elasticity of demand, inelastic demand, elastic demand, demand curve, demand estimation, managerial economics, APEC 3002, applied microeconomics, advertising and sales, regression coefficients, confidence interval, 95% confidence level