Turn in all of the following problems.

**Problem 1:**- For simple linear regression, show that . This fact has an interesting geometric interpretation, and will be proven later using matrix methods.
**Problem 2:**- The data frame
`faithful`in the library`MASS`has 272 observations on two paired variables,`eruptions`and`waiting`. A value of`eruptions`represents the duration (in minutes) of an eruption of the Old Faithful geyser, and the corresponding value of`waiting`gives the waiting time until the next eruption. Suppose that the an eruption of duration 4 minutes and 30 seconds has just been observed. Compute a 95% prediction interval for the waiting time until the next eruption. What assumptions are needed to justify this interval? Can you think of any likely violations of these assumptions? Do residual plots reveal anything? **Rawlings:**- # 1.9 (this can be done by hand if you wish), 1.10,
1.17.

Thu Apr 18 15:52:37 EDT 1996