A researcher is interested in the relationship between highway mileage and weight of vehicles.

Using the below Excel Summary Output, what percentage of the Gas Mileage is explained by the Weight of the Car?

Excel SUMMARY OUTPUT

Regression Statistics
Multiple R 0.739110566
R Square
Adjusted R Square
Standard Error 2.863911526
Observations 30

ANOVA
df SS MS F
Regression 1 276.5109683 276.511 33.71267
Residual 28 229.6556984 8.201989
Total 29 506.1666667

Coefficients Standard Error t Stat P-value
Intercept 46.32975138 3.739022985 12.39087 6.97E-13
Curb Weight -0.00603459 0.001039325 -5.80626 3.09E-06

Question

A researcher is interested in the relationship between highway mileage and weight of vehicles.

Using the below Excel Summary Output, what percentage of the Gas Mileage is explained by the Weight of the Car?

Excel SUMMARY OUTPUT				
				
Regression Statistics			
Multiple R	0.739110566			
R Square				
Adjusted R Square				
Standard Error	2.863911526			
Observations	30

ANOVA				
 	df	SS	MS	F
Regression	1	276.5109683	276.511	33.71267
Residual	28	229.6556984	8.201989	
Total	29	506.1666667	 	 
				
 	Coefficients	Standard Error	t Stat	P-value
Intercept	46.32975138	3.739022985	12.39087	6.97E-13
Curb Weight	-0.00603459	0.001039325	-5.80626	3.09E-06

BlackTom AI · Accepted Answer

To interpret how much of the variation in Gas Mileage is explained by the Car Weight, we look at the R-squared value, which represents the proportion of total variance accounted for by the regression model.
Option 1: 73.91% — This number corresponds to the given Multiple R value of 0.7391, which is the square root of R-squared. While 0.7391 indicates a strong correlation, it is not the percent of variance explained; R-squared would be the square of this, giving a notably smaller percentage.
Option 2: 54.62% — This aligns with the calculation from the ANOVA table: R-squared = Regression SS / Total SS = 276.511 / 506.167 ≈ 0.5462, which is 54.62%. This is the standard measure of variance explained by the predictor weight in the model.
Option 3: 45.37% — This value is close to 1 minus 0.5462, which would represent the proportion of unexplained variance (1 − R-squared). It does not reflect the amount explained, so it misinterprets what R-squared conveys.
Option 4: 46.32% — This figure resembles the rounded form of 0.4632, but it does not match the actual R-squared derived from the given SS values. It would indicate a different proportion if the totals or sums of squares were inconsistent, which they are not here.
Option 5: 33.71% — This number would occur if a smaller portion of variance were explained by the model, which is not supported by the SS totals provided. The correct ratio from the data clearly differs from this value.
In summary, the key idea is that the R-squared value, computed from the regression SS relative to the total SS, determines the percentage of variance in Gas Mileage explained by Weight. Among the options, 54.62% matches that calculation exactly, while the other choices reflect common misreads: confusing R with R-squared, misinterpreting explained vs. unexplained variance, or rounding errors that do not fit the underlying ANOVA numbers.

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