Chapter 11.3, Problem 16Q

Regression techniques can be very useful in situations where one variable—say, $y$—is difficult to measure but $x$ is not. Once such an $xy$-relationship has been “calibrated,” based on a set of $\left(x_i,y_i\right)$’s, future values of $Y$ can be easily estimated using ${\hat{\beta }}_0+{\hat{\beta }}_{1}x$. Determining the volume of an irregularly shaped object, for example, is often difficult, but weighing that object is likely to be easy. The following table shows the weights (in kilograms) and the volumes (in cubic decimeters) of eighteen children between the ages of five and eight $\left(15\right)$. The estimated regression line has the equation $y=-0.104+0.988x$, where $s=0.202$.

• (a) Construct a 95% confidence interval for $E\left(Y|14.0\right)$.

• (b) Construct a 95% prediction interval for the volume of a child weighing 14.0 kilograms.

Weight, $x$	Volume, $y$	Weight, $x$	Volume, $y$
17.1	16.7	15.8	15.2
10.5	10.4	15.1	14.8
13.8	13.5	12.1	11.9
15.7	15.7	18.4	18.3
11.9	11.6	17.1	16.7
10.4	10.2	16.7	16.6
15.0	14.5	16.5	15.9
16.0	15.8	15.1	15.1
17.8	17.6	15.1	14.5