7 a graph airplane count y versus helicopter count x

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Statistics for Management and Economics + XLSTAT Bind-in
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Chapter 18 / Exercise 18.41
Statistics for Management and Economics + XLSTAT Bind-in
Keller
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Statistics for Management and Economics + XLSTAT Bind-in
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Chapter 18 / Exercise 18.41
Statistics for Management and Economics + XLSTAT Bind-in
Keller
Expert Verified
(a) Graph airplane count ( Y ) versus helicopter count ( X ), and draw in the estimated regress line. (b) Carry out two t -tests regarding the slope β 1 : the usual test of H 0 : β 1 and also a test of H 0 : β 1 = 1. Which test seems more relevant to this study? (c) Run the following code in R : lm1=lm(air~heli) new=data.frame(heli=seq(28, 88, 0.5)) pred.PI = predict(lm1, new, level=0.95, interval="prediction") pred.CI = predict(lm1, new, level=0.95, interval="confidence") n=10; x.mean=mean(heli); sxx=sum((heli-x.mean)^2); half.width=summary(lm1)$sigma*sqrt(2*qf(0.95, 2, n-2))* sqrt(1/n+(new$heli-x.mean)^2/sxx) band.CI=cbind(pred.CI[,1]-half.width, pred.CI[,1]+half.width) plot(c(28, 88), range(air, pred.PI, pred.CI, band.CI), type="n") points(heli, air, xlab="Manatee Counts from Helicopter", ylab="From Airplane") abline(lm1) lines(new$heli, pred.PI[,2], lty=1, col="red") lines(new$heli, pred.PI[,3], lty=1, col="red") lines(new$heli, pred.CI[,2], lty=2, col="blue") lines(new$heli, pred.CI[,3], lty=2, col="blue") lines(new$heli, band.CI[,1], lty=3, col="green") lines(new$heli, band.CI[,2], lty=3, col="green") Explain what are plotted in the figure (you don’t need to include this figure in your homework). Explain why the blue intervals are shorter than the red intervals. Explain why the blue intervals are shorter than the green intervals. (d) If the helicopter count really were accurate, and airplane observers counted no imaginary manatees (although they might miss some real ones), the relation between these two counts should be a regression through the origin (because when X = 0, we should have Y = 0 too). Conduct a regression of airplane count on helicopter count by excluding the intercept, and graph the result. Is the slope in this graph significantly different from 1? 14. Old Faithful Geyser Data. The data set faithful gives information about eruptions of the Old Faithful geyser in Yellowstone National Park, Wyoming, USA. Variables are eruptions : the eruption time in minutes; waiting : the waiting time to the next eruption in minutes. Fit a linear regression model to predict waiting from eruptions . (a) What’re the estimated slope and intercept? (b) Give an interpretation of the estimated slope. Construct a 99% CI for the slope. (c) Report the residual standard error and the R 2 . 8
(d) Display the fitted regression line on the scatter plot of the data points. Remember to add proper labels for the x and y coordinates. (e) Construct a 95% interval estimate for the average waiting time to the next eruption for individuals who arrive at the end of an eruption which lasts 250 seconds. (f) Mike has just arrived at the end of an eruption which lasted 250 seconds. Give a 95% interval estimate for the time Mike will have to wait for the next eruption. ( Hint : you first need to explain the estimated interval in this case is a PI, instead of CI.) 15. Continue with the Old Faithful Geyser Data. From the scatter plot, we can see that the data form two clusters . We divide the data into two groups based on eruptions < 3 or not. Fit a linear regression model for each group. (a) Report the estimated slope and intercept for each group. (b) Display the fitted regression lines on the scatter plot of the data points. You should use different line types for these two lines, and also add a legend to explain what the two line types mean. (c) Report the corresponding R 2 for each group. Compare them with the R 2 from the regression model using all the data. Any explanation for such a big discrepancy? Hint

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