A recent article in BusinessWeek listed the “Best Small Companies.” We are inter

Question

A recent article in BusinessWeek listed the “Best Small Companies.” We are interested in the current results of the companies' sales and earnings. A random sample of 12 companies was selected and the sales and earnings, in millions of dollars, are reported below.

Let sales be the independent variable and earnings be the dependent variable.

(a) Compute the correlation coefficient. (Round your answer to 3 decimal places.) Coefficient of correlation

(b) Compute the coefficient of determination. (Round your answer to 3 decimal places.) Coefficient of determination

(c) Determine the regression equation. (Round your answers to 4 decimal places.)

(d) For a small company with $70 million in sales, estimate the earnings. (Round your answer to 4 decimal places.) Estimated earnings

Sales (S millions) $18.6 12.4 15.1 54.7 25.1 68.6 Sales Earnings (S millions) (S millions) Company Earnings (S millions) Company Papa John's International$87.4 Applied Innovation Integracare Wall Data Davidson & Associates Chico's FAS $4.5 Checkmate Electronics $ 4.6 16.5 16.3 70.0 59.8 49.4 4.4 Royal Grip 1.2 M-Wave 7.9 Serving-N-Slide 6.5 Daig 4.4 Cobra Golf 1.9 3.4 8.2 5.8 12.7

Explanation / Answer

Open R or R Studio and run the following commands:

sales=c(87.4,16.5,16.3,70.0,59.8,49.4,18.6,12.4,15.1,54.7,25.1,68.6)
earnings=c(4.5,4.4,1.2,7.9,6.5,4.4,4.6,1.9,3.4,8.2,5.8,12.7)
se=data.frame(sales,earnings)
se_reg=lm(earnings~sales,data = se)
se_reg_summary=summary(se_reg)

a) cor(sales,earnings) #gives coefficient of correlation

Answer : 0.631

b) se_reg_summary$r.squared #coefficient of determination

Answer : 0.398

c)   se_reg

Output :

Call:
lm(formula = earnings ~ sales, data = se)

Coefficients:
(Intercept) sales
2.43519 0.07345   

Coefficients(rounding to 4 decimal places):
(Intercept) sales
2.4352 0.0735   

Therefore, the regression equation is : earnings = 0.0735*sales + 2.4352 (Answer)

d) if sales = $70 million then estimated earnings = 0.0735*70 + 2.4352 (put sales = 70 in the above equation)

= 7.5802

therefore, estimated earnings would be $7.5802 million (Answer)

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