Chapter 4, Problem 5UTA

The data in this section are taken from this source:

Based on King. Cuchlaine A. M. Physical Geography. Oxford: Basil Blackwell.

Throughout the world, natural ocean beaches are beautiful sights to see. If you have visited natural beaches, you may have noticed that when the gradient or dropoff is steep, the grains of sand tend to be larger. In fact, a man-made beach with the "wrong” size granules of sand tends to be washed away and eventually replaced when the proper size grain is selected by the action of the ocean and the gradient of the bottom. Since man-made beaches are expensive, grain size is an important consideration.

In the data that follow $,x=$ median diameter (in millimeters) of granules of sand, and y = gradient of beach slope in degrees on natural ocean beaches.

x	y
0.17	0.63
0.19	0.70
0.22	0.82
0.235	0.88
0.235	1.15
0.30	1.50
0.35	4.40
0.42	7.30
0.85	11.30

Suppose you have a truckload of sifted sand in which the median size of granules is 0.38 mm. If you want to put this sand on a beach and you don't want the sand to wash away, then what does the least-squares line predict for the angle of the beach? Note: Heavy storms that produce abnormal waves may also wash out the sand. However, in the long run, the size of sand granules that remain on the beach or that are brought back to the beach by long-term wave action

are determined to a large extent by the angle at which the beach drops off. What range of angles should the beach have if we want to be $90\%$ confident that we are matching the size of our sand granules $\left(0.38mm\right)$ to the proper angle of the beach?