Concordia Marketing Research Past Paper (Fall 2024) PDF

MARK 302 Marketing Research Section B Instructor: Dr. Yuyan Wei Fall 2024  Chapter 14 More Powerful Statistical Methods: Regression Analysis Week 9 Tests for relational hypothesis Nominal / Ordinal / Interval / Ordinal...

MARK 302 Marketing Research Section B Instructor: Dr. Yuyan Wei Fall 2024  Chapter 14 More Powerful Statistical Methods: Regression Analysis Week 9 Tests for relational hypothesis Nominal / Ordinal / Interval / Ordinal Interval/ Ratio Ratio Chi-Square Spearman Pearson Z-Test Regression Test Correlation Correlation Nature of regression analysis Bivariate regression analysis is used to analyze the relationship between one independent / predictor variable (X) and the dependent variable (Y) Multiple regression analysis is used to analyze the relationship between two or more independent / predictor variables (X1, X2, X3…) and the dependent variable (Y) Regression analysis ≠ X causes Y X happens before Y ➔ by theory/observation, not regression X & Y are correlated ➔ proved by regression analysis No alternative explanation ➔ partially proved by regression Linear relationship A Salesperson has a base monthly salary of $2500. In addition to that, every time the salesperson sells one unit of product, he gets a commission of $100. The more units he sells, the higher his total salary. Linear equation: Y = a + bX y Total Total salary (Y) = $2500 + $100 x unit sold (X) Salary Intercept: a = $2500 $100 Slope: b = $100 $2,500 x Units sold Regression line Least-squares estimation procedure finds a straight line that best fits the actual observations. y ෡ +𝒆 ෝ + 𝒃𝑿 𝒀=𝒂 Total Salary Y = dependent variable (total salary) 𝑎ො = estimated intercept (base salary) 𝑏෠ = estimated slope / coefficient x (commission per unit sold) Units sold X = independent variable (unit sold) e = error (difference between actual value and predicted value) Regression analysis – interpret results 𝒀=𝒂 ෡ +𝒆 ෝ + 𝒃𝑿 𝑌 = 2418 + 102.2 𝑋 + 𝑒 y Total Salary ෝ = 2418 𝒂 Base salary is $2418 when units sold is 0 ෡ = 102.2 x 𝒃 Units Commission is $102.2 per unit sold sold Regression analysis – interpret results ෡ +𝒆 ෝ + 𝒃𝑿 𝒀=𝒂 y Hypotheses: Total Salary H0: 𝑏෠ = 0 (no relationship between X & Y) Ha: 𝑏෠ ≠ 0 (there is a relationship between X & Y) The Simple Regression statistical test reports x 𝒂 ෡ ෝ&𝒃 Units sold Probability of the computed F-statistic (p-value) R2 / adjusted R2: percentage of the variance in Y that is explained by X F-statistic If p-value 0.05 ➔ fail to reject H0 at 95% Step 3: R2 = 0.015 Airtime explains only 1% of variance in monthly bill Step 4: face validity (see next example for details)  Step 3: Variance Explained Variations in monthly phone Variations in airtime bill explained by not used in explaining the error term variation in monthly phone bill Monthly Airtime Phone Bill Variations in airtime used R2 in explaining variation in monthly phone bill R2 = 0.015 Bivariate Regression: Is one variable enough? Notice the example model above did not explain a lot of the variance in monthly phone bill More than one X variable can help to increase the amount of variance explained Potential omitted variable bias Thus, multiple regression is needed to have a more comprehensive understanding of the phenomena Multiple regression analysis ෡𝟏𝑿𝟏 + 𝒃 ෝ+𝒃 𝒀=𝒂 ෡𝟐𝑿𝟐 + ⋯ 𝒃 ෡𝒏𝑿𝒏 + 𝒆 Y = dependent variable 𝑎ො = estimated intercept 𝑏෠ i = estimated slope / coefficient of Xi Xi = independent variable e = error (difference between actual value and predicted value) Hypotheses: H0: 𝑏෠ 1 = 𝑏෠ 2…… = 𝑏෠ n = 0 no relationship between independent variables and dependent variable Ha: at least one 𝑏෠ i ≠ 0 there is a relationship between at least one independent variable and dependent variable)  Multiple regression analysis – interpret results X1 = Airtime Y = Monthly phone bill X2 = Data usage ෝ = 38.99 𝒂 ෡𝟏 = 0.09 𝒃 ෡𝟐 = 0.15 𝒃 Step 1: Equation: Monthly bill (Y) = 38.99 + 0.09 x airtime (X1) + 0.15 x data usage (X2) Step 3: P-value of t-stats 1) The baseline monthly bill is 38.99; 2) holding data usage constant, 1 unit increase P-value of 𝑏෠ 1 = 0.08 > 0.05 in airtime links to 0.09 unit increase in bill; 3) holding airtime constant, 1 unit ➔ Can’t reject 𝑏෠ 1 = 0 at 95% increase in data usage links to 0.15 unit increase in bill. Step 2: P-value of F-stats = 0.000 ➔ reject H0: 𝑏 ̂1 = 𝑏 ̂2…… = 𝑏 ̂n = 0 at 95% P-value of 𝑏෠ 2 < 0.000 significance level ➔ Reject 𝑏෠ 2 = 0 at 95% Step 4: Adjusted R2 = 0.09 ➔ airtime and data usage together explain 9% of variance in monthly bill  Interpret results – face validity Step 5: We also need to think about Face Validity of the intercept & coefficient(s) even when p-value of their t-tests are significant ➔ does the sign (+ or -) make sense in reality? Example 1: Y = 38.99 + 0.09 X1 + 0.15 X2 Y = monthly phone bill, X1 = air time, X2 = data usage Both coefficients are positive ➔ monthly bill increases when air time / data usage increase ➔ make sense Example 2: Y = 38.99 + 0.09 X1 + 0.15 X2 Y = sales, X1 = price, X2 = advertising Coefficient of advertising is 0.15 (+) ➔ sales increases when advertising increases ➔ make sense Coefficient of price is 0.09 (+) ➔ sales increases when price increases? Doesn’t make sense unless you are studying luxury products In-class exercise #1 Below are regression results that show the effect of price and advertising on sales for an energy drink. Interpret the results following the 5 steps we discussed earlier. In-class exercise #1 solution Step 1: Equation: sales (Y) = 1175.08 - 980.49 x price (X1) + 178.65 x advertising (X2) Intercept = 1175.08 (baseline sales); 1 unit increase in price, associated with 980.49 unit decrease in sales; 1 unit increase in ads, associated with 178.65 unit crease in sales. Step 2: P-value of F-stats = 0.000 ➔ reject H0: 𝑏 ̂1 = 𝑏 ̂2 = 0 at 99% significance level Step 3: P-value of t-stats Price : P = 0.0001 < 0.01 ➔ reject 𝑏 ̂1 = 0 at 99% Advertising: P = 0.0049 < 0.01 ➔ reject 𝑏 ̂2 = 0 at 99% Step 4: Adjusted R2 = 0.59 ➔ price and advertising together explain 59% of variance in the monthly bill, which is considered substantial. Step 5: face validity Price has a negative effect on sales ➔ make sense in reality Ad has a positive effect on sales ➔ make sense in reality In-class exercise #2 The operation manager of FastTransport Company is considering multiple regression analysis with number of cargos per shipment and weight of cargo as independent variables and transportation costs as the dependent variable. The manager has devised the following regression equation: Y = a + b1X1 + b2X2 = 23.15 – 4.07 X1 + 21.46 X2 where X1 is the number of cargos per shipment and X2 is the weight of the cargo. a. Interpret a, b1 and b2. b. Interpret the face validity of this regression analysis In-class exercise #2 solution a. The terms respectively represent: The intercept – baseline transportation costs = 23.15. Holding the size of the cargo constant, The equation indicates that a decrease of $4.07 in cost can be expected with each additional cargo. Holding the number of cargo constant, the equation indicates that an increase of $21.46 in cost can be expected with a unit increase in the weight of the cargo. b. The baseline transportation cost is positive, which makes sense. In addition, b2 = -4.07 means the number of cargo is negatively related to costs, which may not make much sense since the more cargo shipped, the higher the total transportation cost it may involve (e.g. maintenance, management, and space cost per cargo). When the cargo weight increases, the cost should also increase. Therefore, the regression analysis seems appropriate.