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 are in millions of dollars. They found the following regression equation.
Sales = 250 + 6.75 TV + 3.5 Radio + 2.3 Magazine
One of following interpretations 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.
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.
Explanation: The other answers are incorrect because they do not accurately reflect the information given in the regression equation. The equation states that for every million dollars spent on TV, sales increase by $6.75 million, after allowing for the effects of the other kinds of advertising.
The correct interpretation is option 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.
Let's go through each option and explain what's wrong with the others:
a) If they did no advertising, their income would be $250 million.
This interpretation is incorrect because the constant term in the regression equation, 250, represents the estimated sales when all advertising variables (TV, radio, magazine) are set to zero. It does not represent the household appliance manufacturer's income without any advertising.
b) Every million dollars spent on radio makes sales increase $3.5 million, all other things being equal.
This interpretation is incorrect because the coefficient for the radio variable in the regression equation is 3.5. It shows that for every additional unit of advertising spent on radio (in millions), the sales increase by 3.5 units (in millions). The interpretation stating that for every million dollars spent on radio, sales increase by $3.5 million is incorrect.
c) Every million dollars spent on magazines increases TV spending $2.3 million.
This interpretation is incorrect because the regression equation represents the relationship between sales and the three primary means of advertising - TV, radio, and magazine. It does not provide any information about the relationship between spending on magazines and the increase in TV spending.
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.
This interpretation is correct. The coefficient for the TV variable in the regression equation is 6.75. It indicates that for every additional unit of advertising spent on TV (in millions), the sales increase by 6.75 units (in millions), while accounting for the effects of the other advertising variables. Hence, this interpretation accurately represents the relationship between TV advertising spending and sales.
The correct interpretation is 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.
Explanation:
a) If they did no advertising, their income would be $250 million.
This interpretation is incorrect because the constant term (250) represents the base sales without any advertising. It does not necessarily mean that the household appliance manufacturer's income would be $250 million if they did no advertising. The constant term is just an intercept value and should not be interpreted in this way.
b) Every million dollars spent on radio makes sales increase $3.5 million, all other things being equal.
This interpretation is incorrect because the coefficient for the variable "Radio" (3.5) represents the change in sales for each unit increase in Radio advertising, while holding all other variables constant. It does not represent the change in sales for every million dollars spent on radio advertising. The coefficient represents the change in sales for each unit increase in the variable, which may or may not be equal to $1 million.
c) Every million dollars spent on magazines increases TV spending $2.3 million.
This interpretation is incorrect because the regression equation does not provide any information about the relationship between magazine spending and TV spending. The coefficient for the variable "Magazine" (2.3) represents the change in sales for each unit increase in Magazine advertising, while holding all other variables constant. It does not provide any information about the relationship between magazine spending and TV spending.
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.
This interpretation is correct because the coefficient for the variable "TV" (6.75) represents the change in sales for each unit increase in TV advertising, while holding all other variables constant. It indicates that, on average, sales increase by about $6.75 million for every million dollars spent on TV advertising, after considering the effects of the other advertising variables.