Problem 2. [SW 14.9] You have a sample of size n = 1 with data y₁ = 2 and x₁ = 1. You are interested in the value of 3 in the regression Y = XB+ u. (Note there is no intercept.) (a) Plot the sum of squared residuals (y₁ - br₁)² as function of b. You can choose your own range for b, one reasonable choice is b € [-2,5]. (Use any software you prefer. Excel is one option.) (b) Show that the least squares estimate of 3 is BOLS = 2. (c) Using ARidge = 1, plot the Ridge penalty term ARidgeb² as a function of b. (d) Using Ridge = 1, plot the Ridge penalized sum of squared residuals (y₁ - bx₁)² + XRidgeb². Please put all three lines in one plot.

Problem 2. [SW 14.9] You have a sample of size n = 1 with data y₁ = 2 and x₁ = 1. You are interested in the value of 3 in the regression Y = XB+ u. (Note there is no intercept.) (a) Plot the sum of squared residuals (y₁ - br₁)² as function of b. You can choose your own range for b, one reasonable choice is b € [-2,5]. (Use any software you prefer. Excel is one option.) (b) Show that the least squares estimate of 3 is BOLS = 2. (c) Using ARidge = 1, plot the Ridge penalty term ARidgeb² as a function of b. (d) Using Ridge = 1, plot the Ridge penalized sum of squared residuals (y₁ - bx₁)² + XRidgeb². Please put all three lines in one plot.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.EA: Extended Application Using Extrapolation To Predict Life Expectancy

Problem 5EA

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Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?
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