The US Federal Trade Commission made measurements on 25 different varieties of cigarettes: tar, nicotine, and carbon monoxide content. These are substances that are considered hazardous. Consider predicting the carbon monoxide (mg) emitted from the tar (mg) and nicotine (g) content. The weight of the cigarette (g) was also measured. The correlations between carbon monoxide (mg) and the explanatory variables are shown below.

Correlation

95% CI for ρ

P-Value

Carbon monoxide content (mg)

Tar content (mg)

0.957

0.905, 0.981

Question

The US Federal Trade Commission made measurements on 25 different varieties of cigarettes: tar, nicotine, and carbon monoxide content.  These are substances that are considered hazardous.  Consider predicting the carbon monoxide (mg) emitted from the tar (mg) and nicotine (g) content.  The weight of the cigarette (g) was also measured. The correlations between carbon monoxide (mg) and the explanatory variables are shown below.

Correlation

95% CI for ρ

P-Value

Carbon monoxide content (mg)

Tar content (mg)

0.957

0.905, 0.981

< 0.001

Carbon monoxide content (mg)

Nicotine content (mg)

0.926

0.837, 0.967

< 0.001

Carbon monoxide content (mg)

Weight (g)

0.464

0.084, 0.726

0.019

Consider fitting a multiple linear regression to these data, with Carbon monoxide content as the response variable and Tar content, Nicotine content and Weight as potential explanatory variables.  Which of the following is true? (Tick all that apply.)

BlackTom AI · Accepted Answer

The question asks us to evaluate several statements about fitting a multiple linear regression model with Carbon monoxide content as the response and Tar content, Nicotine content, and Weight as predictors, given the shown correlations.
Option 1: 'Examining the associations between the explanatory variables is not relevant to interpreting the correlations provided and the results of the multiple regression.' Here, it is important to consider correlations among predictors because multicollinearity can influence the stability and interpretation of regression coefficients. The correlations between Tar, Nicotine, and Weight are relevant when assessing how much unique variance each predictor contributes beyond the others. Therefore, dismissing the associations among predictors as irrelevant is not correct; understanding predictor–predictor relationships is indeed relevant to interpreting a multiple regression model.
Option 2: 'The R2 value for the multiple linear regression model with all three explanatory variables must be at least 91.6%.' Recall that when you add predictors to a regression, R-squared cannot decrease; it either stays the same or increases. The model with Tar content alone would yield R2 = (correlation between CO and Tar)^2 ≈ 0.957^2 ≈ 0.916. Since adding Nicotine content and Weight cannot reduce explained variance, the model with all three predictors must have R2 at least as large as this value, i.e., at least about 0.916 or 91.6%. This statement aligns with the general property of R2 in multiple regression, assuming no data issues. 
Option 3: 'The regression coefficients won’t make sense as tar and nicotine content are measured in mg and weight is measured in grams.' The concern here is about unit differences. In regression, coefficients reflect the effect of a one-unit change in the predictor, so differences in units affect the scale of coefficients but not their validity or interpretability in principle. Coefficients can be interpreted in their respective units or after standardization; thus asserting that coefficients 'won’t make sense' because of differing units is not accurate. This statement misinterprets how regression interprets coefficients and how units influence magnitude rather than meaning.
Option 4: 'All three explanatory variables have statistically significant correlations with the response variable, so it follows that they must each be statistically significant in a multiple regression model when all three are included in the model.' A significant simple correlation with the response does not guarantee significance in the presence of other predictors. In multiple regression, a predictor can lose statistical significance if its shared variance with the response is largely captured by other predictors (multicollinearity or overlapping information). Therefore, this blanket implication is incorrect.

MAST90007_2025_WIN 2025 Quiz 3

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