Question

In: Statistics and Probability

The data in the table represent the weights of various domestic cars and their miles per...

The data in the table represent the weights of various domestic cars and their miles per gallon in the city for the 2008 model year. For the data from the first 11​ cars, the​ least-squares regression line is y=−0.0062x+42.4755. A twelfth car weighs 2,705 pounds and gets 14 miles per gallon. Compute the coefficient of determination of the expanded data set​ (including the twelfth​ car). What effect does the addition of the twelfth car to the data set have on Rsquared2​?

Car   Weight_(pounds)_x   Miles_per_Gallon_y
1   3765   21
2   3980   19
3   3532   22
4   3174   21
5   2582   27
6   3729   18
7   2601   26
8   3775   18
9   3313   19
10   2991   26
11   2753   26
12   2705   14

1) The coefficient of determination of the expanded data set is

2)How does the addition of the twelfth car to the data set affect Rsquared2​? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

a) It increases by __

b) It decreases by __

c) it does not affect it

Solutions

Expert Solution

Solution: We can use the excel regression data analysis tool to find the answer to the given questions. The excel output is given below:

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.4988
R Square 0.2488
Adjusted R Square 0.1737
Standard Error 3.7269
Observations 12
ANOVA
df SS MS F Significance F
Regression 1 46.01604742 46.01604742 3.312875614 0.098755827
Residual 10 138.9006192 13.89006192
Total 11 184.9166667
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 34.3965 7.2120 4.7694 0.0008 18.3272 50.4658
Weight_(pounds)_x   -0.0040 0.0022 -1.8201 0.0988 -0.0089 0.0009

1) The coefficient of determination of the expanded data set is

Answer:

2)How does the addition of the twelfth car to the data set affect Rsquared2​? Select the correct choice below​ and, if​necessary, fill in the answer box to complete your choice.

b) It decreases by 0.5365

Explanation:

R-square for 11 pairs of data set is 0.7853 and the R-square for the 12 pairs of data set is 0.2488. Therefore, the R-square is decreased by 0.7853-0.2488=0.5365


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