Chapter 3.4, Problem 15E

Remainder Round all answers to two decimal places unless otherwise indicated.

The Effect of Sampling Error on Linear Regression A stream that feeds a lake is flooding, and during this flooding period, the depth of water in the lake is increasing. The actual depth of the water at a certain point in the lake is given by the linear function $D=0.8t+52$ feet, where t is measured in hours since the flooding began. A hydrologist does not have this function available and is trying to determine experimentally how the water level is rising. She sits in a boat and, each half-hour, drops a weighted line into the water to measure the depth to the bottom. The motion of the boat and the waves at the surface make exact measurement impossible. Her compiled data are given in the following table.

$\mathit{t}$ = hours since flooding began	$\mathit{D}=$ measured depth in feet
0	51.9
0.5	52.5
1	52.9
1.5	53.3
2	53.7

1. a.

Plot the data points.

1. b.

Find the equation of the regression line for D as a function of t, and explain in practical terms the meaning of the slope.

1. c.

Add the graph of the regression line to the plot of the data points.

1. d.

Add the graph of the depth function $D=0.8t+52$ to the picture. Does it appear that the hydrologist was able to use her data to make a close approximation of the depth function?

1. e.

What was the actual depth of the water at t = 3 hours?

1. f.

What prediction would the hydrologist’s regression line give for the depth of the water at t = 3?