x y -2 5 2 3 0 1 1 4 1 2 (a) Compute the estimates β0 and β1. (b) Suppose we have a new individual whose X is 0.5. Can you tell me what their Y is? Why or why not? (c) Can you estimate E(Y)? (d) Explain the meaning of β1 and b1. (e) Find the residuals for this model. Show that they add up to 0. (f) Estimate σ^2 for this model. (g) What is the distribution of b1 and b0? What is the distribution of b0 + 3b1? (h) Suppose we have a new individual whose X is 0.5. What is Ŷ for the individual? What is the mean and variance of Ŷ? What is the distribution of Ŷ? (i) Estimate the variance of b1. Create a 95% confidence interval for β1. (j) Test the hypothesis that H0: β1 = 0 vs H1: β1 ≠ 0 at level 0.05
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Problem 13 dealt with how to form a confidence interval for the value of a line at a point $x_{0} .$ Suppose that instead we want to predict the value of a new observation, $Y_{0},$ at $x_{0}$ $$ Y_{0}=\beta_{0}+\beta_{1} x_{0}+e_{0} $$ by the estimate $$ \hat{Y}_{0}=\hat{\beta}_{0}+\hat{\beta}_{1} x_{0} $$ a. Find an expression for the variance of $\hat{Y}_{0}-Y_{0},$ and compare it to the expression for the variance of $\hat{\mu}_{0}$ obtained in part (a) of Problem $13 .$ Assume that $e_{0}$ is independent of the original observations and has the variance $\sigma^{2}$ b. Assuming that $e_{0}$ is normally distributed, find the distribution of $Y_{0}-Y_{0} .$ Use this result to find an interval $I$ such that $P\left(Y_{0} \in I\right)=1-\alpha$. This interval is called a $100(1-\alpha) \%$ prediction interval.
The data from exercise 2 follow. $$\frac{x_{i}|3 \quad 12 \quad 6 \quad 20 \quad 14}{y_{i}|55 \quad 40 \quad 55 \quad 10 \quad 15}$$ $$\begin{array}{l}{\text { a. Compute the mean square error using equation }(12.15) .} \\ {\text { b. Compute the standard error of the estimate using equation }(12.16) .} \\ {\text { c. Compute the estimated standard deviation of } b_{1} \text { using equation }(12.18).} \\ {\text { d. Use the } t \text { test to test the following hypotheses }(\alpha=.05) :}\end{array}$$ $$\begin{array}{c}{H_{0} : \beta_{1}=0} \\ {H_{\mathrm{a}} : \beta_{1} \neq 0}\end{array}$$ $$\begin{array}{l}{\text { e. Use the } F \text { test to test the hypotheses in part (d) at a } .05 \text { level of significance. Present }} \\ {\text { the results in the analysis of variance table format. }}\end{array}$$
Suppose that $X_{1}, X_{2}, \ldots, X_{m}$ and $Y_{1}, Y_{2}, \ldots, Y_{n}$ are independent random samples, with the variables $X_{i}$ normally distributed with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$ and the variables $Y_{i}$ normally distributed with mean $\mu_{2}$ and variance $\sigma_{2}^{2} .$ The difference between the sample means, $\bar{X}-\bar{Y},$ is then a linear combination of $m+n$ normally distributed random variables and, by Theorem $6.3,$ is itself normally distributed. a. Find $E(\bar{X}-\bar{Y})$. b. Find $V(\bar{X}-\bar{Y})$. c. Suppose that $\sigma_{1}^{2}=2, \sigma_{2}^{2}=2.5,$ and $m=n .$ Find the sample sizes so that $(\bar{X}-\bar{Y})$ will be within 1 unit of $\left(\mu_{1}-\mu_{2}\right)$ with probability .95
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