7, (26%) A substance used in biological and medical research is shipped by air-f

Question

7, (26%) A substance used in biological and medical research is shipped by air-freight to users in cartons of 1000 ampules. The data below, involving 10 shipments, were collected on the number of times the carton was transferred from one aircraft to another over the shipment route (X) and the number of ampules found to be broken upon arrivalC). Assume that 1 0 2 0 31 01 2 0 16 9 17 12 22 13 8 15 19 11 (a) (896) Verify that the estimated regression function is Y = 10.2 + 4x . Interpret the y-intercept and the slope in this context (b) (14%) Conduct an F test to decide whether or not there is a linear association between the number of times a carton is transferred and the number of broken ampoules; control risk at 0.05. State the alternatives the test statistic, the critical value, the decision rule, and your conclusion. What is the corresponding p-value? Interpret the value (c) (3%) Obtain the ristatistic for the test in (b) and demonstrate numerically its equivalence to the F statistic obtained in (b). TABLE B.2 Percentiles of the t Distribution. Entry is t(A1) where P { t(v1S t(A; v)) = A .60 10.325 0.727 1.376 1.963 3.078 6.314 12.706 2 0.289 0.6171.061 1.386 1.886 2.920 4.303 3.182 3 0.277 0.584 0.978 1.250 1638 2.353 0.271 0.569 0.941 1.190 1.533 2.132 4 0 0.267 0.559 0.920 1.156 1.476 2.0152.571 6 0.265 0.553 0.906 1.134 1.440 1.943 2.447 0.263 0.549 0.896 1.119 1.415 1.895 9 0.261 0.543 0.883 1.100 1.383 1.833 11 0.260 0.540 0.8761.088 1.363 1.796 0.2620.546 0.8891.108 1.397 1.860 2.306 2 262 8 10 0.2600.42 0.879 1.093 1.3721.812 12 0.259 0.539 0.873 1.083 1.356 2.179 2.160 1.782 13 0259 0.537 0870 1.079 1.350 1.771 14 0.258 0.537 0.868 1.076 1.345 1.761 2.145 15 0.2580.536 0.866 1.074 1.3411.753 2.131

Explanation / Answer

(a) the regression equation y=a+bx=10.2+4x

a=(sum(y)*sum(x2)-sum(x)*sum(xy))/(n*sum(x2)-(sum(x))2)=(142*20-10*182)/(10*20-10*10)=10.2

b=(n*sum(xy)-sum(x)*sum(y))/ (n*sum(x2)-(sum(x))2) =(10*182-142*10)/(10*20-10*10)=4

(b) here p-value of F is less than alpha=0.05 so model is significant at 5% level of significance.

following information has been generated using ms-excel

(c) F(1,8)=72.73 is the square t(8)=8.53

so t2(v)~F(1,v)

Regression Statistics Multiple R 0.949157996 R Square 0.900900901 Adjusted R Square 0.888513514 Standard Error 1.483239697 Observations 10 ANOVA df SS MS F Significance F Regression 1 160 160 72.72727 2.74867E-05 Residual 8 17.6 2.2 Total 9 177.6 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 10.2 0.663324958 15.37707857 3.18E-07 8.670369905 11.7296301 X Variable 1 4 0.469041576 8.528028654 2.75E-05 2.918388187 5.081611813

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