Variation in regression line and coefficient of determination
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Problems 1, 3
1. Use the graph below to describe the total variation about a regression line in words and in symbols.
This is the answer from the book, but how? S(yi - y)2; the sum of the squares of the differences between the y-values of each ordered pair and the mean of the y-values of the ordered pairs.
3 Use the graph to below describe the unexplained variation about a regression line in words and in symbols.
This is the answer from the book, but how? S(yi - yi)2; the sum of the squares of the differences between the observed y-values and the predicted y-values.
Problems 11, 13, 15
For #11, 13, 15 Find the (a) coefficient of determination and interpret the results, and (b) standard error of estimate se and interpret the results.
11 Retail Space and Sales ? The following table represents the total square footage (in billions) of retailing space at shopping centers and their sales (in billions of U.S. dollars) for 11 years. The equation of the regression line is
Total square footage, x 1.6 2.3 3.0 3.4 3.9 4.6
Sales, y 123.2 211.5 385.5 475.1 641.1 716.9
Total square footage, x 4.7 4.8 4.9 5.0 5.1
Sales, y 768.2 806.6 851.3 893.8 933.9
13. Earnings of Men and Women ? The following table represents median weekly earnings (in U.S. dollars) of full-time male and female workers for five years. The equation of the regression line is
Median weekly earnings of male workers,x 312 419 485 538 557
Median weekly earnings of female workers,y 201 290 348 406 418
15. Campaign Money ? The money raised and spent (both in millions of U.S. dollars) by all congressional campaigns for eight recent years are shown in the table. The data can be modeled by the regression equation = 1.020x - 25.854.
Money raised,x 354.7 397.2 472.0 477.6
Money spent,y 342.4 374.1 450.9 459.0
Money raised,x 471.7 659.3 740.5 790.5
Money spent,y 446.3 680.2 725.2 765.3
Problem 17
Constructing and Interpreting Prediction Intervals: In the next question construct the indicated prediction interval and interpret the results.
The number of initial public offerings of stock issued in a recent 12-year period and the total proceeds of these offerings (in millions of U.S. dollars) are listed below.
No. of issues,x 332 694 518 222 209 172
Proceeds,y 6284.8 17,738.8 16,745.7 6111.7 6082.0 4519.0
No. of issues,x 366 512 667 571 575 865
Proceeds,y 16,283.2 23,379.8 34,461.1 22,771.9 29,270.8 48,789.8
17. Proceeds? Construct a 95% prediction interval for the proceeds from initial public offerings when the number of issues is 712.
Problem 29
Use the information given below to answer the question:
29 Coefficient of Determination ? Find the coefficient of determination. What can you conclude?
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Solution Summary
The expert examines variation in regression lines and coefficients of determinations.
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The detailed step by step solutions are given in the attached solutions file.
Answer to Question (1):
Total variation about a regression line is a measure of variation of the Yi values around their mean .
Therefore, Total Sum of Squares = SST =
Answer to Question (2):
Total variation (or total sum of squares, SST) can be subdivided into explained variation (or Regression Sum of Squares, SSR ), that which is attributable to the relationship between X and Y, and unexplained variation (or Error Sum of Squares, SSE ) that which is attributable to factors other than relationship between X and Y.
Therefore, Explained variation = SSR =
Unexplained variation = SSE =
Solution to Question (3):
To find the coefficient of determination and the standard error of estimate, the required computations are done in the following table.
Square footage Sales Fitted Value Residual
x y xy y2 ŷ e e2
1.6 123.2 197.12 15178.24 79.48 43.72 1911.438
2.3 211.5 486.45 44732.25 241.04 -29.54 872.6116
3.0 385.5 1156.50 148610.25 402.9 -17.4 302.76
3.4 475.1 1615.34 225720.01 494.92 -19.82 392.8324
3.9 641.1 2500.29 411009.21 610.32 30.78 947.4084
4.6 716.9 3297.74 513945.61 771.88 -54.98 3022.8
4.7 768.2 3610.54 590131.24 794.96 -26.76 716.0976
4.8 806.6 3871.68 650603.56 818.04 -11.44 130.8736
4.9 851.3 4171.37 724711.69 841.12 10.18 103.6324
5.0 893.8 4469.00 798878.44 864.2 29.6 876.16
5.1 933.9 4762.89 872169.21 887.28 46.62 2173.424
SUM = 43.3 6807.1 30138.92 4995689.71 6806.14 11450.04
Solution to (a):
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