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8. Consider the model $y = \alpha + \beta_1 x_1 + \beta_2 x_2 + \epsilon$ We employed the least squares method using 25 observations. This lead us to obtain the following results (just for clarification, $cov(a, b_1, b_2)$ is a variance-covariance matrix for estimates (i.e., $a, b_1, b_2$). For instance, 0.019195 is covariance between a and $b_1$. $\begin{bmatrix} a \\ b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} 0.96587 \\ 0.69914 \\ 1.7769 \end{bmatrix}$ $cov(a, b_1, b_2) = \begin{bmatrix} 0.21812 & 0.019195 & -0.050301 \\ 0.019195 & 0.048526 & -0.031223 \\ -0.050301 & -0.031223 & 0.037120 \end{bmatrix}$ $\hat{\sigma}^2 = 3.041$ $R^2 = 0.9311$ 8-A) compute the total variation, unexplained variation and explained variation for the above model (5pts) 8-B) compute 95% interval estimates for $\beta_1$ and $\beta_2$ (3pts) 8-C) use a t-test to conduct the hypothesis test specified as: H0: $\beta_1 = 1$ against H1: $\beta_1 < 1$ (4pts) 8-D) test the joint hypothesis H0: $\beta_1 = 1, \beta_2 = 0$; you may use your answers in part (8-A) to avoid uncessary time-consuming calculation procedures. (5pts) 8-E) test the hypothesis H0: $2\beta_1 = \beta_2$ (6pts)

          8. Consider the model
$y = \alpha + \beta_1 x_1 + \beta_2 x_2 + \epsilon$
We employed the least squares method using 25 observations. This lead us to obtain the following
results (just for clarification, $cov(a, b_1, b_2)$ is a variance-covariance matrix for estimates (i.e., $a, b_1, b_2$).
For instance, 0.019195 is covariance between a and $b_1$.
$\begin{bmatrix} a \\ b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} 0.96587 \\ 0.69914 \\ 1.7769 \end{bmatrix}$
$cov(a, b_1, b_2) = \begin{bmatrix} 0.21812 & 0.019195 & -0.050301 \\ 0.019195 & 0.048526 & -0.031223 \\ -0.050301 & -0.031223 & 0.037120 \end{bmatrix}$
$\hat{\sigma}^2 = 3.041$  $R^2 = 0.9311$
8-A) compute the total variation, unexplained variation and explained variation for the above model
(5pts)
8-B) compute 95% interval estimates for $\beta_1$ and $\beta_2$
(3pts)
8-C) use a t-test to conduct the hypothesis test specified as: H0: $\beta_1 = 1$ against H1: $\beta_1 < 1$ (4pts)
8-D) test the joint hypothesis H0: $\beta_1 = 1, \beta_2 = 0$; you may use your answers in part (8-A) to avoid
uncessary time-consuming calculation procedures.
(5pts)
8-E) test the hypothesis H0: $2\beta_1 = \beta_2$
(6pts)

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8 consider the model y alpha beta1 x1 beta2 x2 epsilon we employed the least squares method using 25 observations this lead us to obtain the following results just for clarification cova 19783

Added by Daniel P.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Dominador Tan

03/15/2023

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The total variation is the total sum of squares (SST), the unexplained variation is the sum of squared errors (SSE), and the explained variation is the sum of squares due to regression (SSR). We are given that $R^2 = 0.9311$ and $\hat{\sigma}^2 = 3.041$. The Show more…

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8. Consider the model $y = \alpha + \beta_1 x_1 + \beta_2 x_2 + \epsilon$ We employed the least squares method using 25 observations. This lead us to obtain the following results (just for clarification, $cov(a, b_1, b_2)$ is a variance-covariance matrix for estimates (i.e., $a, b_1, b_2$). For instance, 0.019195 is covariance between a and $b_1$. $\begin{bmatrix} a \\ b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} 0.96587 \\ 0.69914 \\ 1.7769 \end{bmatrix}$ $cov(a, b_1, b_2) = \begin{bmatrix} 0.21812 & 0.019195 & -0.050301 \\ 0.019195 & 0.048526 & -0.031223 \\ -0.050301 & -0.031223 & 0.037120 \end{bmatrix}$ $\hat{\sigma}^2 = 3.041$ $R^2 = 0.9311$ 8-A) compute the total variation, unexplained variation and explained variation for the above model (5pts) 8-B) compute 95% interval estimates for $\beta_1$ and $\beta_2$ (3pts) 8-C) use a t-test to conduct the hypothesis test specified as: H0: $\beta_1 = 1$ against H1: $\beta_1 < 1$ (4pts) 8-D) test the joint hypothesis H0: $\beta_1 = 1, \beta_2 = 0$; you may use your answers in part (8-A) to avoid uncessary time-consuming calculation procedures. (5pts) 8-E) test the hypothesis H0: $2\beta_1 = \beta_2$ (6pts)

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Transcript

00:02 Good day everyone.

00:03 So here in our solution for part a, first one is we need to solve the test statistic.

00:10 So we have f is equals to msr, divided msc.

00:15 So we have equal to 7 ,026 .1 divided by 161 .56.

00:22 So we have equal to 43 .4976 or equal to 43 .50.

00:31 Next one is to find a degree of freedom due to the regression.

00:35 So we have equal to p and this would be equal to two.

00:40 Next one is the degree of freedom due to the error.

00:43 So we have df.

00:46 That is the denominator which is equals to n minus p minus one.

00:54 So we have equal to 10 minus 2 minus 1 which is equals to 7.

00:58 So our level of significance is 0 .05.

01:02 Now for the f critical value, so basically here, using the f distribution tables for numerator decrease of freedom 2, we have f that is 0 .052 and 7, which is equals to 4 .74.

01:20 So the test statistic value is greater than the critical value.

01:24 So what we're going to do here is we need to reject the null hypothesis.

01:34 Therefore, it can be concluded that a significant relationship is present between y and two independent variables, so that is x sub 1 and x sub 2.

01:43 Now for part b, we need to solve the t test statistics.

01:49 So we have t is equals to b1, divide divide s, b1.

01:53 So we have equal to 2 .01 divided 0 .2471...

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