Chapter 9: Problem 5
Assume that the sample \(\left(x_{1}, Y_{1}\right), \ldots,\left(x_{n},
Y_{n}\right)\) follows the linear model \((9.6 .1)\). Suppose \(Y_{0}\) is a future
observation at \(x=x_{0}-\bar{x}\) and we want to determine a predictive
interval for it. Assume that the model \((9.6 .1)\) holds for \(Y_{0}\); i.e.,
\(Y_{0}\) has a \(N\left(\alpha+\beta\left(x_{0}-\bar{x}\right),
\sigma^{2}\right)\) distribution. We will use \(\hat{\eta}_{0}\) of Exercise \(9.6
.4\) as our prediction of \(Y_{0}\)
(a) Obtain the distribution of \(Y_{0}-\hat{\eta}_{0}\). Use the fact that the
future observation \(Y_{0}\) is independent of the sample \(\left(x_{1},
Y_{1}\right), \ldots,\left(x_{n}, Y_{n}\right)\)
(b) Determine a \(t\) -statistic with numerator \(Y_{0}-\hat{\eta}_{0}\).
(c) Now beginning with \(1-\alpha=P\left[-t_{\alpha / 2, n-2}
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.