4. To show your work in this question, please give the equations for the request

Question

4. To show your work in this question, please give the equations for the requested result in matrix-vector format. You do not need to show the expanded matrices or the arithmetic for the answers. A data set contains 50 observations. There are 3 explanatory variables: A, B, and C. Use the following results: 1.3000-0.0200 -0.0400 -0.0100 (x'x)-100200 00500 0.0030 -0.0020 0.0400-0.00300.00340.0010 -0.0100 0.0020 0.0010 0.0047 467.0 b -12.4 3.45 -6.32 MSE = 324 (a) Give the 95% confidence interval for ßi (the coefficient for A). (b) Test Ho : As = 0 vs. Ha : 3 0. In other words, determine whether variable C significantly improves the prediction of Y when variables A and B are already in the el. Report the test statistic for the hypothesis test with degrees of freedom and the associated p-value, and make a decision to accept or reject the null hypothesis. (e) Give a 95% confidence interval for the mean (expected) response when A-40, B C 50. 20 (d) Give a 95% prediction interval for a single response when A-40, B-20, c

Explanation / Answer

a)Var() = 2 (X’X)-1
Here, 2 = 324, n= 50.
Let (X’X)-1 = ((a ij))
Then, Var(j) = 2.a jj
Thus, S.E (j) = Var(j) = . a jj
Hence, S.E(1)= 4.024922
95% CI for 1 : ( 1 ± t 0.025; n-4. SE(1)) = (-12.4 ± 2.012896*4.024922) = (-20.50175 , -4.298251)

b) T-statistic for the test : T = 3/S.E(3)
Hence, S.E(3) = 1.234018
Thus, T = -5.121481
Critical value = t 0.025; n-4 = 2.012896
Since |T| > critical value, we reject H0 and conclude that variable C is significant for the prediction of Y.

c) Estimate = 467-12.4*40+3.45*20-6.32*50 = -276

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