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Chapter 12: Q4E (page 722)

Question: Suppose you fit the first-order multiple regression model y=β0+β1x1+β2x2+ε to n=25 data points and obtain the prediction equationy^=6.4+3.1x1+0.92x2 . The estimated standard deviations of the sampling distributions of β1 and β2are 2.3 and .27, respectively

Short Answer

Expert verified

(a) Do not reject the null hypothesis, hence the value of β1=0

(b) Reject the null hypothesis at 95% significance level, thus, the value of β2=0

(c) 90% confidence interval for β1 is (0.0732, 6.1268).

(d) 99% confidence interval for β2 is (0.16751, 1.67249).

Step by step solution

01

Step-by-Step Solution  Step 1: Testing the significance β1

02

Testing the significance of  β2

Therefore, value of β2≠0

03

Confidence interval for β1

90% confidence interval forβ1is β^2±t0.005sβ2

Therefore, the confidence interval is 3.1±1.316×2.3

Thus, confidence interval forβ1is (0.0732, 6.1268). Here, with 90% accuracy it can be concluded that the value ofβ1will lie between 0.0732 and 6.1268. The value of also falls in the intervals which is a positive sign.

04

Confidence interval for  β2

99% confidence interval forβ2isβ^2±t0.005sβ2β^2±t0.005sβ2

Therefore, the confidence interval is 0.92±2.787×0.27

Thus, confidence interval for β2 is (0.16751, 1.67249). Here, with 99% accuracy it can be concluded that the value of β2will lie between 0.16751 and 1.67249. The value of β^2 also falls in the intervals which is a positive sign

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