Just as you are about to estimate a regression( due tomorrow), massive sunspots

Question

Just as you are about to estimate a regression( due tomorrow), massive sunspots cause magnetic interference that ruins all electrically powered machines (e.g., computers). Instead of giving up and flunking, you decide to calculate estimates from your data ( on per capita income in thousands of U.S. dollars as a function of the percent of the labor force in agriculture in 10 developed countries) using methods like those used in Section 2.1 ( Yi=B0+ B1Xi + Ei) without using a computer. Your data are :

Country A B C D E F G H I J

Per Capita Income 6 8 8 7 7 12 9 8 9 10

% in Agriculture 9 10 8 7 10 4 5 5 6 7

a.) Calculate Beta-hat zero and Beta-hat one.

b.) Calculate R2( r-squared) and R`2 (adjusted r-squared)

c.) If the percent of the labor force in agriculture in another developed country was 8 percent, what level of per capita income ( in thousands of U.S. dollars) would you guess that country had?

Explanation / Answer

a)

average of per capita income,Ybar = (6+8+8+7+7+12+9+8+9+10)/10 = $8.4

average of agriculture, Xbar = (9+10+8+7+10+4+5+5+6+7)/10 = 7.1

B1 = SIGMA(Xi-Xbar)(Yi-Ybar)/SIGMA(Xi-Xbar)^2

SIGMA(Xi-Xbar)(Yi-Ybar)= (9-7.1)*(6-8.4)+(10-7.1)*(8-8.4).................(7-7.1)*(10-8.4) = -22.4

SIGMA(Xi-Xbar)^2 = (9-7.1)^2 + (10-7.1)^2..........................(7-7.1)^2 = 40.9

B1 = -22.4/40.9 = -0.547

B0 = Ybar-B1*Xbar = 8.4+.547*7.1 = 12.284

b)

Rsq = 1-(ycap-ybar)^2/(yi-ybar)^2 = .464

adjusted R^2 = 1-(1-R^2)*(n-1)/(n-k-1) = .3977

where n = 10, k = 2

c)

Ycap = 12.284-.547*8 = 7.908

SUMMARY OUTPUT Regression Statistics Multiple R 0.681686 R Square 0.464696 Adjusted R Square 0.397783 Standard Error 1.329099 Observations 10 ANOVA df SS MS F Significance F Regression 1 12.26797 12.26797 6.944775 0.029931 Residual 8 14.13203 1.766504 Total 9 26.4 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 12.28851 1.534242 8.009498 4.33E-05 8.75054 15.82648 8.75054 15.82648 agri labor -0.54768 0.207824 -2.63529 0.029931 -1.02692 -0.06843 -1.02692 -0.06843

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