Q: Given the regression equation $$Y = 6.577X + 3.584Radio - 2.343Magazine$$, where $$Y is $$total sales in millions, $$X is TV $$advertising in millions, Radio is radio advertising in millions, and Magazine is magazine advertising in millions, what is the predicted sales if a company spends $$5$$ million on TV, $$2$$ million on radio, and $$1$$ million on magazine advertising?

$$Y = 6.577 \times 5 + 3.584 \times 2 - 2.343 \times 1 = 32.885 + 7.168 - 2.343 = 37.71$$ million

Q: Given the regression equation $$Y = a + bX$$, where $$Y is $$the mortgage amount and $$X is $$the interest rate, if the regression output gives $$a = 52.527$$ and $$b = -6.877$$, what is the predicted mortgage amount when the interest rate is $$7.0\%$$?

$$Y = 52.527 - 6.877 \times 7 = 52.527 - 48.139 = 4.388$$ trillion dollars

Q: If the interest rate increases to $$13\%$$, what is the predicted mortgage amount using the regression equation $$Y = 52.527 - 6.877X$$?

$$Y = 52.527 - 6.877 \times 13 = 52.527 - 89.401 = -36.874$$ trillion dollars

Q: Given the regression equation $$Y = 240 + 6.75TV + 3.5Radio - 2.34Magazine$$, which of the following is a correct interpretation if a company spends $$10$$ million on TV, $$5$$ million on radio, and $$2$$ million on magazine advertising?

Total sales would be $$240 + 6.75 \times 10 + 3.5 \times 5 - 2.34 \times 2 = 240 + 67.5 + 17.5 - 4.68 = 320.32$$ million.

Question 1

Given the regression equation $$Y = 6.577X + 3.584Radio - 2.343Magazine$$, where $$Y is $$total sales in millions, $$X is TV $$advertising in millions, Radio is radio advertising in millions, and Magazine is magazine advertising in millions, what is the predicted sales if a company spends $$5$$ million on TV, $$2$$ million on radio, and $$1$$ million on magazine advertising?

Accepted Answer

$$Y = 6.577 	imes 5 + 3.584 	imes 2 - 2.343 	imes 1 = 32.885 + 7.168 - 2.343 = 37.71$$ million

Question 2

A scatterplot of mortgage amounts (in trillions of dollars) vs. interest rate in the US shows a correlation $$r = -0.845. $$What does this value of $$r$$ indicate?

Accepted Answer

There is a strong negative linear relationship between mortgage amounts and interest rates.

Question 3

Given the regression equation $$Y = a + bX$$, where $$Y is $$the mortgage amount and $$X is $$the interest rate, if the regression output gives $$a = 52.527$$ and $$b = -6.877$$, what is the predicted mortgage amount when the interest rate is $$7.0\%$$?

Accepted Answer

$$Y = 52.527 - 6.877 	imes 7 = 52.527 - 48.139 = 4.388$$ trillion dollars

Question 4

If the interest rate increases to $$13\%$$, what is the predicted mortgage amount using the regression equation $$Y = 52.527 - 6.877X$$?

Accepted Answer

$$Y = 52.527 - 6.877 	imes 13 = 52.527 - 89.401 = -36.874$$ trillion dollars

Question 5

What does the coefficient $$b in $$the regression equation $$Y = a + bX$$ represent in the context of mortgage amounts and interest rates?

Accepted Answer

The change in mortgage amount for each one percentage point increase in interest rate.

Question 6

If both mortgage amount and interest rate are standardized (converted to $$z-$$scores), what would be the regression equation predicting standardized mortgage amount from standardized interest rate?

Accepted Answer

$$z_Y = r 	imes z_X$$

Question 7

A group of statistics students surveyed 200 students: 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors. What sampling design did they use?

Accepted Answer

Stratified random sampling

Question 8

Researchers found that men who died of myocardial infarction (heart attack) tended to be shorter than men who did not. Is this an experiment?

Accepted Answer

No, this is an observational study.

Question 9

Is it correct to conclude that shorter men are at higher risk of dying from a heart attack based on the study described above?

Accepted Answer

No, because correlation does not imply causation.

Question 10

In a regression predicting hill race record times from climb (meters), the regression output gives $$R^2 = 0.82$. What does this value mean?

Accepted Answer

82\% of the variation in record times is explained by climb.

Question 11

Given the regression equation $$Y = 240 + 6.75TV + 3.5Radio - 2.34Magazine$$, which of the following is a correct interpretation if a company spends $$10$$ million on TV, $$5$$ million on radio, and $$2$$ million on magazine advertising?

Accepted Answer

Total sales would be $$240 + 6.75 	imes 10 + 3.5 	imes 5 - 2.34 	imes 2 = 240 + 67.5 + 17.5 - 4.68 = 320.32$$ million.

Question 12

If the regression equation for predicting mortgage amount from interest rate is $$Y = 52.527 - 6.877X$$, what is the mortgage amount when the interest rate is $$0\%$$?

Accepted Answer

$$Y = 52.527$$ trillion dollars

Question 13

What does the intercept $$a in $$the regression equation $$Y = a + bX$$ represent in the context of predicting mortgage amount from interest rate?

Accepted Answer

The predicted mortgage amount when the interest rate is zero.

Question 14

A regression output for hill race record times includes a coefficient for Climb of $$0.015. $$How should this coefficient be interpreted?

Accepted Answer

For each additional meter of climb, the record time increases by $$0.015$$ minutes.

Question 15

If a regression equation is $$Y = 240 + 6.75TV + 3.5Radio - 2.34Magazine$$, what is the effect on sales if radio advertising increases by $$1$$ million dollars, holding other variables constant?

Accepted Answer

Sales increase by $$3.5$$ million dollars.

Question 16

Which of the following is a plausible reason for standardizing variables before running a regression?

Accepted Answer

To compare the relative importance of predictors measured on different scales.

Question 17

A regression equation for predicting sales from advertising expenditures is $$Y = 220 + 6.75TV + 3.5Radio - 2.34Magazine. If a $$company does no advertising, what is the predicted sales?

Accepted Answer

$$220$$ million

Question 18

In a regression analysis, what does it mean if the $$p-$$value for a predictor variable is less than $$0.05$$?

Accepted Answer

The predictor variable is statistically significant at the $$5\%$$ level.

Statistics Study Guide: Regression, Correlation, and Data Analysis

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