CH14 exercises solutions

CH14 exercises solutions - X X X X X X X X X X X X X X X X...

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X X X X X X X X X X X X X X X X X AP Statistics Solutions to Packet 14 X Inference for Regression Inference about the Model Predictions and Conditions X X X X X X X X X X X X X X
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2 HW #32 1, 2, 6, 7 14.1 AN EXTINCT BEAST, I Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Here are the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five fossil specimens that preserve both bones: Femur: Humerus: 38 41 56 63 59 70 64 72 74 84 The strong linear relationship between the lengths of the two bones helped persuade scientists that all five specimens belong to the same species. (a) Examine the data. Make a scatterplot with femur length as the explanatory variable. Use your calculator to obtain the correlation r and the equation of the least-squares regression line. Do you think the femur length will allow good prediction of humerus length? (b) Explain in words what the slope β of the true regression line says about Archaeopteryx. What is the estimate of β from the data? What is your estimate of the intercept α of the true regression line? β represents how much we can expect the humerus length to increase when femur length increases by 1 cm, b (the estimate of β ) is 1.1969, and the estimate of α is a = 3.660. (c) Calculate the residuals for the five data points. Check that their sum is 0 (up to roundoff error.) Use the residuals to estimate the standard deviation σ in the regression model. You have now estimated all three parameters in the model. The residuals are 0.8226, 0.3668, 3.0425, 0.9420, and 0.9110; the sum is 0.0001 (but carrying a different number of digits might change this). Squaring and summing the residuals gives 11.79, so that 11.79/ 3 1.982. s = = The correlation is r = 0.994, and linear regression gives ˆ 3.66 0 1.1969 yx = - + The scatterplot below shows a strong, positive, linear relationship, which is confirmed by r .
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3 14.2 BACKPACKS Body weights and backpack weights were collected for eight students. Weight (lbs): Backpack weight (lbs): 120 26 187 30 109 26 103 24 131 29 165 35 158 31 116 28 These data were entered into a statistics package and least-squares regression of backpack weight on body weight was requested. Here are the results: Predictor Constant BodyWT Coef 16.265 0.09080 Stdev 3.937 0.02831 t-ratio 4.13 3.21 P 0.006 0.018 s = 2.270 R-sq = 63.2% R-sq(adj) = 57.0% (a) What is the equation of the least-squares line? (Hint: Look for the column “Coef.” What is the intercept? What is the slope? backpack weight = 16.265 + 0.0908(body weight). The intercept is 16.265 and the slope is 0.0908 (b) The model for regression inference has three parameters, which we call α , β , and σ . Can you determine the estimates for α and β from the computer output? What are they? The estimate for α is the intercept of the least-squares line, that is, 16.265. The estimate for β is the slope of the least-squares line, that is, 0.0908.
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CH14 exercises solutions - X X X X X X X X X X X X X X X X...

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