Question

Question: Suppose you fit the first-order multiple regression model $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{\epsilon }$ to n=25 data points and obtain the prediction equation$\hat{\mathbf{y}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{4}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{1}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{92}{\mathbf{x}}_{\mathbf{2}}$ . The estimated standard deviations of the sampling distributions of ${\beta }_1$ and ${\mathit{\beta }}_{\mathbf{2}}$are 2.3 and .27, respectively

Short answer

(a) Do not reject the null hypothesis, hence the value of ${\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}\mathbf{0}$

(b) Reject the null hypothesis at 95% significance level, thus, the value of ${\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

(c) 90% confidence interval for ${\mathit{\beta }}_{\mathbf{1}}$ is (0.0732, 6.1268).

(d) 99% confidence interval for ${\beta }_2$ is (0.16751, 1.67249).

Step-by-step solution

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

Testing the significance of  β2

Therefore, value of ${\mathbf{\beta }}_2\ne \mathbf{0}$

Confidence interval for β1

90% confidence interval for${\beta }_1$is ${\hat{\mathbf{\beta }}}_{\mathbf{2}}\mathbf{\pm }{\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{005}}{\mathbf{s\beta }}_{\mathbf{2}}$

Therefore, the confidence interval is $\mathbf{3}\mathbf{.}\mathbf{1}\mathbf{\pm }\mathbf{1}\mathbf{.}\mathbf{316}\mathbf{\times }\mathbf{2}\mathbf{.}\mathbf{3}$

Thus, confidence interval for${\beta }_1$is (0.0732, 6.1268). Here, with 90% accuracy it can be concluded that the value of${\beta }_1$will lie between 0.0732 and 6.1268. The value of also falls in the intervals which is a positive sign.

Confidence interval for  β2

99% confidence interval for${\mathbf{\beta }}_{\mathbf{2}}$is${\hat{\mathbf{\beta }}}_{\mathbf{2}}\mathbf{\pm }{\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{005}}{\mathbf{s\beta }}_{\mathbf{2}}$${\hat{\mathbf{\beta }}}_{\mathbf{2}}\mathbf{\pm }{\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{005}}{\mathbf{s\beta }}_{\mathbf{2}}$

Therefore, the confidence interval is $\mathbf{0}\mathbf{.}\mathbf{92}\mathbf{\pm }\mathbf{2}\mathbf{.}\mathbf{787}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{27}$

Thus, confidence interval for ${\mathbf{\beta }}_{\mathbf{2}}$ is (0.16751, 1.67249). Here, with 99% accuracy it can be concluded that the value of ${\mathbf{\beta }}_{\mathbf{2}}$will lie between 0.16751 and 1.67249. The value of ${\hat{\mathbf{\beta }}}_{\mathbf{2}}$ also falls in the intervals which is a positive sign