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Homework set #2 (100 points total)
Theory Part I: (30 points)
1. (10 points) Prove that
2. (10 points) Prove that
This has to do with equivalence of the F and t statistics to test Ho: β1=0 vs Ha: β1≠0. Hint:
you can use the equality in (1).
3. (10 points) Prove that, for the simple linear regression of y on x
Applied Part; “body fat” analysis: (70 points)
Please refer to the data description and general guidelines file when performing the following
analyses and preparing your write-up.
A. Consider again the simple linear regression of body fat percentage versus abdomen
circumference, and the simple linear regression of body fat percentage versus weight/height.
For each
•
(10 points) Build and interpret 95% and 99% confidence intervals for the slope.
•
(5 points) Test and interpret β1=0 vs Ha: β1≠0 (you can find the p-values in the
regression output).
(10 points) Does the data contain evidence that, on average, for each additional cm of
abdomen circumference the body fat percentage increases by more than 0.5 points? (you
have to set up and perform a test of hypothesis to answer this question).
B. (5 points) For both the regression of body fat percentage versus abdomen circumference,
and the simple linear regression for body fat percentage versus weight/height, produce the
fitted line plot with 95% confidence interval for the mean response and 95% prediction
interval “bands” superimposed. Comment and interpret.
(10 points) In addition, compute 95% confidence intervals for mean body fat percentage and
95% prediction intervals for body fat percentage in correspondence of the 10, 25, 50, 75 and
2
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90 percentiles of abdomen circumference, and of weight/height ratio. Comment and
interpret.
C. (10 points) For both the regression of body fat percentage versus abdomen
circumference, and the simple linear regression