In: Economics
With milk sales sagging of late, The Milk Processor Education Program (MPEP) decided to move on from the famous "Got Milk" ad slogan in favor of a new one, "Milk Life." The new tagline emphasizes milk's nutritional benefits, including its protein content. MPEP began collecting data on the number of gallons of milk households consumed weekly (in millions), weekly price per gallon, and weekly expenditures on milk advertising (in hundreds of dollars) for the period following the launch of the new campaign. These data, in forms to estimate both a linear model and log-linear model, are available via the link below. Use these data to perform two regressions: a linear regression and a log-linear regression.
Excel Data File
Which model does a better job fitting the data?
The --------------- model.
Suppose that the weekly price of milk is $3.40 per gallon and MPEP
decides to ramp up weekly advertising by 35 percent to $150 (in
hundreds). Use the best-fitting regression model to estimate the
weekly quantity of milk consumed after this advertising
increase.
Instructions: Round your intermediate calculations
and enter your response rounded to three decimal places.
million gallons per week
Linear Model | Log-Linear Model | |||||
Q | P | A | lnQ | lnP | lnA | |
4.76 | 2.46 | 472.68 | 1.56 | 0.90 | 6.16 | |
0.90 | 4.28 | 326.41 | -0.10 | 1.45 | 5.79 | |
1.74 | 3.72 | 357.36 | 0.55 | 1.31 | 5.88 | |
0.96 | 4.20 | 475.82 | -0.04 | 1.43 | 6.17 | |
2.38 | 4.14 | 494.25 | 0.87 | 1.42 | 6.20 | |
1.28 | 4.59 | 458.62 | 0.25 | 1.52 | 6.13 | |
2.86 | 3.30 | 421.67 | 1.05 | 1.19 | 6.04 | |
1.87 | 4.34 | 534.85 | 0.63 | 1.47 | 6.28 | |
2.19 | 3.31 | 524.75 | 0.78 | 1.20 | 6.26 | |
1.38 | 3.35 | 370.35 | 0.32 | 1.21 | 5.91 | |
0.21 | 4.53 | 420.16 | -1.54 | 1.51 | 6.04 | |
3.55 | 2.63 | 333.79 | 1.27 | 0.97 | 5.81 | |
2.44 | 4.40 | 437.32 | 0.89 | 1.48 | 6.08 | |
1.94 | 4.36 | 442.70 | 0.66 | 1.47 | 6.09 | |
2.50 | 3.24 | 375.67 | 0.91 | 1.18 | 5.93 | |
2.92 | 3.45 | 546.36 | 1.07 | 1.24 | 6.30 | |
4.94 | 2.97 | 391.17 | 1.60 | 1.09 | 5.97 | |
2.14 | 3.22 | 498.00 | 0.76 | 1.17 | 6.21 | |
3.89 | 3.34 | 530.17 | 1.36 | 1.20 | 6.27 | |
6.91 | 2.24 | 527.36 | 1.93 | 0.81 | 6.27 | |
3.41 | 4.04 | 440.93 | 1.23 | 1.40 | 6.09 | |
1.16 | 4.10 | 480.35 | 0.15 | 1.41 | 6.17 | |
1.60 | 3.99 | 404.91 | 0.47 | 1.38 | 6.00 | |
4.09 | 3.22 | 512.00 | 1.41 | 1.17 | 6.24 | |
2.69 | 2.98 | 346.29 | 0.99 | 1.09 | 5.85 | |
2.41 | 4.30 | 383.47 | 0.88 | 1.46 | 5.95 | |
2.25 | 2.84 | 434.26 | 0.81 | 1.04 | 6.07 | |
2.48 | 3.96 | 548.37 | 0.91 | 1.38 | 6.31 | |
3.79 | 2.49 | 357.71 | 1.33 | 0.91 | 5.88 | |
3.33 | 3.29 | 445.73 | 1.20 | 1.19 | 6.10 | |
2.61 | 4.02 | 524.55 | 0.96 | 1.39 | 6.26 | |
2.40 | 4.05 | 487.87 | 0.88 | 1.40 | 6.19 | |
3.92 | 2.46 | 343.13 | 1.37 | 0.90 | 5.84 | |
3.42 | 3.45 | 353.81 | 1.23 | 1.24 | 5.87 | |
0.80 | 3.40 | 334.47 | -0.23 | 1.22 | 5.81 | |
5.79 | 2.95 | 330.57 | 1.76 | 1.08 | 5.80 | |
3.58 | 2.69 | 363.91 | 1.28 | 0.99 | 5.90 | |
1.58 | 3.79 | 383.71 | 0.46 | 1.33 | 5.95 | |
1.14 | 3.37 | 430.37 | 0.13 | 1.21 | 6.06 | |
1.04 | 4.64 | 501.84 | 0.04 | 1.54 | 6.22 | |
4.88 | 2.66 | 447.12 | 1.59 | 0.98 | 6.10 | |
4.31 | 2.25 | 404.38 | 1.46 | 0.81 | 6.00 | |
2.23 | 3.94 | 449.29 | 0.80 | 1.37 | 6.11 | |
1.38 | 4.42 | 327.99 | 0.32 | 1.49 | 5.79 | |
1.62 | 3.13 | 332.39 | 0.49 | 1.14 | 5.81 | |
1.38 | 4.45 | 450.16 | 0.33 | 1.49 | 6.11 | |
6.20 | 2.38 | 467.40 | 1.82 | 0.87 | 6.15 | |
4.17 | 3.69 | 528.60 | 1.43 | 1.31 | 6.27 | |
4.08 | 4.02 | 533.73 | 1.41 | 1.39 | 6.28 | |
0.08 | 4.30 | 355.81 | -2.55 | 1.46 | 5.87 | |
3.82 | 2.80 | 462.42 | 1.34 | 1.03 | 6.14 | |
1.17 | 4.51 | 549.78 | 0.16 | 1.51 | 6.31 | |
3.26 | 2.42 | 366.63 | 1.18 | 0.88 | 5.90 | |
2.44 | 4.37 | 429.74 | 0.89 | 1.47 | 6.06 | |
4.16 | 2.53 | 399.57 | 1.42 | 0.93 | 5.99 | |
2.63 | 3.63 | 521.95 | 0.97 | 1.29 | 6.26 | |
4.94 | 2.80 | 356.59 | 1.60 | 1.03 | 5.88 | |
1.84 | 4.36 | 416.24 | 0.61 | 1.47 | 6.03 | |
4.71 | 3.12 | 435.99 | 1.55 | 1.14 | 6.08 | |
6.46 | 2.40 | 464.62 | 1.87 | 0.87 | 6.14 | |
2.79 | 3.51 | 353.37 | 1.03 | 1.25 | 5.87 | |
4.09 | 3.07 | 425.12 | 1.41 | 1.12 | 6.05 | |
4.76 | 2.32 | 481.72 | 1.56 | 0.84 | 6.18 | |
3.05 | 3.45 | 376.30 | 1.12 | 1.24 | 5.93 | |
0.87 | 4.44 | 536.86 | -0.13 | 1.49 | 6.29 | |
3.12 | 2.50 | 493.52 | 1.14 | 0.92 | 6.20 | |
1.34 | 3.11 | 454.69 | 0.29 | 1.13 | 6.12 | |
1.93 | 3.24 | 487.07 | 0.66 | 1.17 | 6.19 | |
1.64 | 2.87 | 461.69 | 0.50 | 1.05 | 6.13 | |
4.39 | 2.97 | 410.84 | 1.48 | 1.09 | 6.02 | |
5.76 | 2.33 | 480.66 | 1.75 | 0.84 | 6.18 | |
4.40 | 2.82 | 381.62 | 1.48 | 1.04 | 5.94 | |
6.22 | 3.14 | 456.97 | 1.83 | 1.14 | 6.12 | |
1.10 | 3.89 | 461.39 | 0.09 | 1.36 | 6.13 | |
4.12 | 2.67 | 430.43 | 1.42 | 0.98 | 6.06 | |
5.40 | 2.73 | 438.53 | 1.69 | 1.01 | 6.08 | |
2.75 | 4.52 | 336.00 | 1.01 | 1.51 | 5.82 | |
5.12 | 2.28 | 519.90 | 1.63 | 0.83 | 6.25 | |
3.94 | 3.25 | 536.25 | 1.37 | 1.18 | 6.28 | |
5.69 | 2.18 | 439.75 | 1.74 | 0.78 | 6.09 | |
0.44 | 4.27 | 352.57 | -0.82 | 1.45 | 5.87 | |
1.89 | 3.62 | 397.69 | 0.64 | 1.29 | 5.99 | |
4.02 | 3.32 | 345.17 | 1.39 | 1.20 | 5.84 | |
3.70 | 3.43 | 507.56 | 1.31 | 1.23 | 6.23 | |
3.26 | 2.43 | 330.67 | 1.18 | 0.89 | 5.80 | |
2.98 | 2.97 | 433.20 | 1.09 | 1.09 | 6.07 | |
2.09 | 4.32 | 462.14 | 0.74 | 1.46 | 6.14 | |
5.68 | 2.25 | 515.33 | 1.74 | 0.81 | 6.24 | |
4.33 | 2.65 | 508.14 | 1.47 | 0.98 | 6.23 | |
4.97 | 3.63 | 510.41 | 1.60 | 1.29 | 6.24 | |
2.89 | 3.60 | 343.16 | 1.06 | 1.28 | 5.84 | |
2.25 | 3.37 | 365.82 | 0.81 | 1.22 | 5.90 | |
0.17 | 3.77 | 425.56 | -1.79 | 1.33 | 6.05 | |
3.96 | 2.87 | 347.36 | 1.38 | 1.06 | 5.85 | |
4.08 | 2.97 | 326.06 | 1.40 | 1.09 | 5.79 | |
3.49 | 3.94 | 527.12 | 1.25 | 1.37 | 6.27 | |
4.21 | 4.10 | 475.28 | 1.44 | 1.41 | 6.16 | |
2.25 | 4.09 | 475.69 | 0.81 | 1.41 | 6.16 | |
2.40 | 3.93 | 536.42 | 0.88 | 1.37 | 6.28 | |
1.61 | 4.10 | 325.89 | 0.48 | 1.41 | 5.79 |
SUMMARY OUTPUT for linear model | ||||||||
Regression Statistics | ||||||||
Multiple R | 0.740 | |||||||
R Square | 0.547 | |||||||
Adjusted R Square | 0.538 | |||||||
Standard Error | 1.063 | |||||||
Observations | 100 | |||||||
ANOVA | ||||||||
df | SS | MS | F | Significance F | ||||
Regression | 2 | 132.51 | 66.26 | 58.61 | 0.00 | |||
Residual | 97 | 109.66 | 1.13 | |||||
Total | 99 | 242.17 | ||||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |
Intercept | 6.520 | 0.823 | 7.921 | 0.000 | 4.886 | 8.153 | 4.886 | 8.153 |
P | -1.614 | 0.151 | -10.658 | 0.000 | -1.915 | -1.314 | -1.915 | -1.314 |
A | 0.005 | 0.002 | 2.962 | 0.004 | 0.002 | 0.008 | 0.002 | 0.008 |
SUMMARY OUTPUT for log-linear model | ||||||||
Regression Statistics | ||||||||
Multiple R | 0.634 | |||||||
R Square | 0.401 | |||||||
Adjusted R Square | 0.389 | |||||||
Standard Error | 0.587 | |||||||
Observations | 100 | |||||||
ANOVA | ||||||||
df | SS | MS | F | Significance F | ||||
Regression | 2 | 22.403 | 11.201 | 32.522 | 0.000 | |||
Residual | 97 | 33.409 | 0.344 | |||||
Total | 99 | 55.812 | ||||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |
Intercept | -1.989 | 2.243 | -0.886 | 0.378 | -6.441 | 2.464 | -6.441 | 2.464 |
lnP | -2.170 | 0.276 | -7.858 | 0.000 | -2.717 | -1.622 | -2.717 | -1.622 |
lnA | 0.911 | 0.370 | 2.459 | 0.016 | 0.176 | 1.646 | 0.176 | 1.646 |
linear model | log-linear model | |
regression equation | ||
0.547 | 0.401 | |
0.538 | 0.389 | |
F-statistic | 58.61 | 32.522 |
For the linear regression model, the estimates indicate that R2 = .547, or that 54.7 percent of the variability in the quantity demanded is explained by price and advertising. In contrast, the R2 for the log-linear model is 0.401, indicating that only 40.1 percent of the variability in the natural log of quantity is explained by variation in the natural log of price and the natural log of advertising. Therefore, the linear regression model appears to do a better job explaining variation in the dependent variable.
This conclusion is further supported by comparing the adjusted R2s and the F-statistics in the two models, which are higher for the linear model compared to log-linear model
If weekly price of milk is $3.40 per gallon and MPEP decides to ramp up weekly advertising by 35 percent to $150