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Chapter 28: Problem 12

A household appliance manufacturer wants to analyze the relationship between total sales and the company's three primary means of advertising (television, magazines, and radio). All values were in millions of dollars. They found the regression equation $$\text { Sales }=250+6.75 \mathrm{TV}+3.5 \text { Radio }+2.3 \text { Magazines.}$$ One of the interpretations below is correct. Which is it? Explain what's wrong with the others. a) If they did no advertising, their income would be \(\$ 250\) million. b) Every million dollars spent on radio makes sales increase \(\$ 3.5\) million, all other things being equal. c) Every million dollars spent on magazines increases TV spending \(\$ 2.3\) million. d) Sales increase on average about \(\$ 6.75\) million for each million spent on TV, after allowing for the effects of the other kinds of advertising.

Short Answer

Expert verified
Correct interpretation: a, b, d; c is incorrect as it misinterprets the relationship.

Step by step solution

01

Analyze Option (a) Interpretation

This statement refers to the intercept of the regression equation. In the equation \( \text{Sales} = 250 + 6.75 \cdot \text{TV} + 3.5 \cdot \text{Radio} + 2.3 \cdot \text{Magazines} \), the intercept is 250. It represents the expected sales in millions when all advertising expenditure is zero. Thus, option a is correct.
02

Analyze Option (b) Interpretation

This option describes the coefficient for the `Radio` advertising variable. The coefficient 3.5 indicates that for every additional million dollars spent on radio advertising, sales increase by 3.5 million dollars, assuming TV and Magazine advertising remain constant. This interpretation is correct.
03

Analyze Option (c) Interpretation

This option incorrectly associates the coefficient for magazines with TV spending, which is not represented in the equation. The coefficient for magazines represents how sales, not TV spending, would change with spending in magazines. Therefore, this interpretation is wrong.
04

Analyze Option (d) Interpretation

This option correctly interprets the coefficient 6.75 for TV advertising. It means that sales increase by 6.75 million dollars for each additional million dollars spent on TV advertising, assuming radio and magazine advertising expenditures remain constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Advertising Effectiveness
When a company invests in advertising, understanding its effectiveness is crucial. Advertising effectiveness quantifies how well advertising campaigns drive sales and impact consumer behavior. In our scenario, incorporating regression analysis helps evaluate the impact of each advertising medium, i.e., television, radio, and magazines.

The regression equation provided, \[ ext{Sales} = 250 + 6.75 imes ext{TV} + 3.5 imes ext{Radio} + 2.3 imes ext{Magazines} \] gives the relationship between sales and advertising expenditures. Each coefficient reflects the average change in sales for an additional million dollars spent on that specific advertising medium.

With this insight, businesses can allocate their advertising budgets more effectively. They can identify which medium yields the highest sales return.

For instance:
  • Spending on TV influences sales significantly, increasing returns by \(6.75\) million for every million spent.
  • Radio advertising increases sales by \(3.5\) million per additional million spent.
  • Magazine advertising boosts sales by \(2.3\) million for the same spend.
By comparing these values, companies can strategically decide based on the highest returns to optimize advertising strategies.
Sales Forecasting
Sales forecasting is about predicting future sales based on historical data. Regression analysis plays a crucial role here, allowing businesses to forecast sales effectively with advertising spending as a key predictor. This involves examining how different variables predict sales outcomes.

In the exercise, the regression equation forecasts sales based on expenditures in TV, radio, and magazines. Using this model, one can predict future sales by plugging in expected spending amounts into the equation.

This predictive model helps in:
  • Budget Optimization: Allocate funds to the most cost-effective advertising channels.
  • Strategic Planning: Set realistic sales targets and prepare for different sales scenarios.
  • Resource Allocation: Plan inventory and supply chain logistics based on predicted sales figures.
This method of sales forecasting thus not only serves to predict future sales but also aids in creating data-driven business strategies that can respond dynamically to changing market conditions.
Interpretation of Coefficients
Understanding coefficients in a regression model is essential for information extraction and decision making. Each coefficient represents the rate of change in the dependent variable (sales) per change in an independent variable (advertising expenditures).

In our regression model, coefficients have particular meanings:
  • The intercept, \(250\), is the baseline sales value when no advertising occurs, indicating potential brand strength or market share without promotional efforts.
  • The coefficient for TV (\(6.75\)) implies for each million dollars spent on TV advertising, sales increase by \(6.75\) million, keeping other spending consistent.
  • The coefficient for Radio (\(3.5\)) shows a \(3.5\) million increase in sales per million spent on radio.
  • The Magazines coefficient (\(2.3\)) suggests sales rise by \(2.3\) million with each additional million spent.
Correctly interpreting these coefficients allows businesses to understand where their advertising spend will be most effective, helping them prioritize investments that maximize returns and drive sales growth.

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Most popular questions from this chapter

What can predict how much a motion picture will make? We have data on a number of movies that includes the USGross (in \(\$$ ), the Budget (\$), the Run Time (minutes), and the average number of Stars awarded by reviewers. The first several entries in the data table look like this: $$\begin{array}{l|c|c|c|c} & \text { USGross } & \text { Budget } & \text { Run Iime } & \\\\\text { Movie } & (\$ \mathrm{M}) & (\$ \mathrm{M}) & \text { (minutes) } & \text { Stars } \\\\\hline \text { White Noise } & 56.094360 & 30 & 101 & 2 \\\\\text { Coach Carter } & 67.264877 & 45 & 136 & 3 \\\\\text { Elektra } & 24.409722 & 65 & 100 & 2 \\\\\text { Racing Stripes } & 49.772522 & 30 & 110 & 3 \\\\\text { Assault on Precinct 13 } & 20.040895 & 30 & 109 & 3 \\\\\text { Are We There Yet? } & 82.674398 & 20 & 94 & 2 \\\\\text { Alone in the Dark } & 5.178569 & 20 & 96 & 1.5 \\\\\text { Indigo } & 51.100486 & 25 & 105 & 3.5\end{array}$$ We want a regression model to predict USGross. Parts of the regression output computed in Excel look like this: $$\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\\\\text { Intercept } & -22.9898 & 25.70 & -0.895 & 0.3729 \\\\\text { Budget(\$) } & 1.13442 & 0.1297 & 8.75 & \leq 0.0001 \\\\\text { Stars } & 24.9724 & 5.884 & 4.24 & \leq 0.0001 \\\\\text { Run Time } & -0.403296 & 0.2513 & -1.60 & 0.1113\end{array}$$ a) Write the multiple regression equation. b) What is the interpretation of the coefficient of \)B u d g e t$ in this regression model?

Chest size might be a good predictor of body fat. Here's a scatterplot of \(\%\)Body Fat vs. Chest Size. A regression of \(\%\)Body Fat on Chest Size gives the following equation: Dependent variable is Pct BF R-squared \(=49.1 \% \quad\) R-squared (adjusted) \(=48.9 \%\) \(s=5.930\) with \(250-2=248\) degrees of freedom \(\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & -52.7122 & 4.654 & -11.3 & <0.0001 \\ \text { Chest Size } & 0.712720 & 0.0461 & 15.5 & <0.0001\end{array}\) a) Is the slope of \(\% B o d y\) Fat on Chest Size statistically distinguishable from 0? (Perform a hypothesis test.) b) What does the answer in part a mean about the relationship between \(\% B o d y\) Fat and Chest Size? We saw before that the slopes of both Waist size and Height are statistically significant when entered into a multiple regression equation. What happens if we add Chest Size to that regression? Here is the output from a regression on all three variables: Dependent variable is Pct BF R-squared \(=72.2 \% \quad\) R-squared (adjusted) \(=71.9 \%\) \(s=4.399\) with \(250-4=246\) degrees of freedom \(\begin{array}{lllccc} & \text { Sum of } & & \text { Mean } & & \\\\\text { Source } & \text { Squares } & \text { df } & \text { Square } & \text { F-Ratio } & \text { P-Value } \\\\\text { Regression } & 12368.9 & 3 & 4122.98 & 213 & <0.0001 \\\\\text { Residual } & 4759.87 & 246 & 19.3491 & &\end{array}\) \(\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & 2.07220 & 7.802 & 0.266 & 0.7908 \\ \text { Waist } & 2.19939 & 0.1675 & 13.1 & <0.0001 \\ \text { Height } & -0.561058 & 0.1094 & -5.13 & <0.0001 \\\ \text { Chest Size } & -0.233531 & 0.0832 & -2.81 & 0.0054\end{array}\) c) Interpret the coefficient for Chest Size. d) Would you consider removing any of the variables from this regression model? Why or why not?

The AFL-CIO has undertaken a study of 30 secretaries' yearly salaries (in thousands of dollars). The organization wants to predict salaries from several other variables. The variables considered to be potential predictors of salary are The variables considered to be potential predictors of salary are \(\mathrm{X} 1=\) months of service \(\mathrm{X} 2=\) years of education \(\mathrm{X} 3=\) score on standardized test \(\mathrm{X} 4=\) words per minute (wpm) typing speed \(\mathrm{X} 5=\) ability to take dictation in words per minute A multiple regression model with all five variables was run on a computer package, resulting in the following output: \(\begin{array}{lccc}\text { Variable } & \text { Coefficient } & \text { Std. Error } & \text { t-Value } \\ \text { Intercept } & 9.788 & 0.377 & 25.960 \\\ \text { X1 } & 0.110 & 0.019 & 5.178 \\ \text { X2 } & 0.053 & 0.038 & 1.369 \\ \text { X3 } & 0.071 & 0.064 & 1.119 \\ \text { X4 } & 0.004 & 0.307 & 0.013 \\ \text { X5 } & 0.065 & 0.038 & 1.734\end{array}\) \(s=0.430 \quad R^{2}=0.863\) Assume that the residual plots show no violations of the conditions for using a linear regression model. a) What is the regression equation? b) From this model, what is the predicted Salary (in thousands of dollars) of a secretary with 10 years (120 months) of experience, 9th grade education (9 years of education), a 50 on the standardized test, 60 wpm typing speed, and the ability to take 30 wpm dictation? c) Test whether the coefficient for words per minute of typing speed \((X 4)\) is significantly different from zero at \(\alpha=0.\) d) How might this model be improved? e) A correlation of Age with Salary finds \(r=0.682,\) and the scatterplot shows a moderately strong positive linear association. However, if \(X 6=A g e\) is added to the multiple regression, the estimated coefficient of \(A g e\) turns out to be \(b_{6}=-0.154 .\) Explain some possible causes for this apparent change of direction in the relationship between age and salary.

How well do exams given during the semester predict performance on the final? One class had three tests during the semester. Computer output of the regression gives Dependent variable is Final \(s=13.46 \quad R-S q=77.7 \% \quad R-S q(a d j)=74.1 \%\) \(\begin{array}{lcccr}\text { Predictor } & \text { Coeff } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & -6.72 & 14.00 & -0.48 & 0.636 \\ \text { Test1 } & 0.2560 & 0.2274 & 1.13 & 0.274 \\\ \text { Test2 } & 0.3912 & 0.2198 & 1.78 & 0.091 \\ \text { Test3 } & 0.9015 & 0.2086 & 4.32 & <0.0001\end{array}\) Analysis of Variance \(\begin{array}{lrcccc}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F-Ratio } & \text { P-Value } \\ \text { Regression } & 3 & 11961.8 & 3987.3 & 22.02 & <0.0001 \\ \text { Error } & 19 & 3440.8 & 181.1 & & \\ \text { Total } & 22 & 15402.6 & & & \end{array}\) a) Write the equation of the regression model. b) How much of the variation in final exam scores is accounted for by the regression model? c) Explain in context what the coefficient of Test3 scores means. d) A student argues that clearly the first exam doesn't help to predict final performance. She suggests that this exam not be given at all. Does Test 1 have no effect on the final exam score? Can you tell from this model? (Hint: Do you think test scores are related to each other?)

The data set on body fat contains 15 body measurements on 250 men from 22 to 81 years old. Is average \%Body Fat related to Weight? Here's a scatterplot: \(\begin{array}{lcccc} \text { Variable } & \text { Coefficient } & \text { SE(Coeff) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & -14.6931 & 2.760 & -5.32 & <0.0001 \\ \text { Weight } & 0.18937 & 0.0153 & 12.4 & <0.0001 \end{array}\) a) Is the coefficient of \(\%\)Body Fat on Weight statistically distinguishable from 0? (Perform a hypothesis test.) b) What does the slope coefficient mean in this regression? We saw before that the slopes of both Waist size and Height are statistically significant when entered into a multiple regression equation. What happens if we add Weight to that regression? Recall that we've already checked the assumptions and conditions for regression on Waist size and Height in the chapter. Here is the output from a regression on all three variables: Dependent variable is Pct BF R-squared \(=72.5 \% \quad\) R-squared (adjusted) \(=72.2 \%\) \(s=4.376\) with \(250-4=246\) degrees of freedom \(\begin{array}{lllll} & \text { Sum of } & & \text { Mean } & \\ \text { Source } & \text { Squares } & \text { df } & \text { Square } & \text { F-Ratio } \\ \text { Regression } & 12418.7 & 3 & 4139.57 & 216 \\ \text { Residual } & 4710.11 & 246 & 19.1468 & \end{array}\) \(\begin{array}{lcccc}\text { Variable } & \text { Coefficient } & \text { SE(Coeft) } & \text { t-Ratio } & \text { P-Value } \\ \text { Intercept } & -31.4830 & 11.54 & -2.73 & 0.0068 \\ \text { Waist } & 2.31848 & 0.1820 & 12.7 & <0.0001 \\ \text { Height } & -0.224932 & 0.1583 & -1.42 & 0.1567 \\\ \text { Weight } & -0.100572 & 0.0310 & -3.25 & 0.0013\end{array}\) c) Interpret the slope for Weight. How can the coefficient for Weight in this model be negative when its coefficient was positive in the simple regression model? d) What does the P-value for Height mean in this regression? (Perform the hypothesis test.)
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