Problem 3-26 Obtain a linear regression line for the data. (Round your intermedi

Question

Problem 3-26

   

Obtain a linear regression line for the data.(Round your intermediate calculations and final answers to 3 decimal places.)

What percentage of the variation is explained by the regression line? (Do not round intermediate calculations. Round your answer to the nearest whole percent. Omit the "%" sign in your response.)

Use the equation determined in part b to predict the expected value of y for x = 42. (Round your intermediate calculations and final answers to 3 decimal places.)

The following data were collected during a study of consumer buying patterns:

Explanation / Answer

b.

b = (N*sum of x*y) - sum of x*sum of y/(n*sum of x^2 - (sum of x)^2)

= (13*31,547-366*1098)/(13*12246-366^2) = 8243/25242 = 0.327

a = (sum of y - b*sum of x)/n = (1098-0.327*366)/13 = 75.268

Thus the regression line: y = 75.268+0.327x

c. percentage of variation being explained by regression line = r^2 or the coefficient of determination.

r = [n*sum of xy - sum of x*sum of y]/[n*sum of x^2 - (sum of x)^2]^1/2*[n*sum of y^2 - (sum of y)^2]^1/2

= [13*31547 - 366*1098]/[13*12246 - 366^2]^1/2*[13*93392 - (1098)^2]^1/2

=[8243]/[158.877]*[92.152] = 0.563013. r^2 = 0.563013^2 = 0.316984

It means that 0.316984*100 = 31.698% of variation is exlained by the regression line.

d. y = 75.268+0.327x (as determined earlier)

putting x as 42 in the above equation:

y = 75.268+0.327*42 = 85.718

Observation X Y x*y x*x 1 18 77 1386 324 2 22 84 1848 484 3 45 79 3555 2025 4 37 83 3071 1369 5 49 96 4704 2401 6 42 91 3822 1764 7 33 82 2706 1089 8 21 80 1680 441 9 10 75 750 100 10 12 75 900 144 11 18 90 1620 324 12 25 91 2275 625 13 34 95 3230 1156 Total 366 1098 31547 12246

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