Chapter 11.4, Problem 9P

In Problems 7-12, parts (a) and (b) relate to testing $\rho$. Part (c) requests the value of $S_e$. Parts (d) and (c) relate to confidence intervals for prediction. Part (f) relates to testing $\beta$.

Scuba Diving: Depth What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the “optimal time” to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let x = depth of dive in meters, and let y = optimal time in hours. A random sample of divers gave the following data (based on information taken from Textbook of Medical Physiology by A. C. Guyton, M.D.):

x	14.1	24.3	30.2	38.3	51.3	20.5	22.7
y	2.58	2.08	1.58	1.03	0.75	2.38	2.20

(a) Verify that ${\displaystyle \sum x}=201.4,{\displaystyle \sum y}=12.6,{\displaystyle \sum x^2}=6735.46,{\displaystyle \sum y^2}=25.607,{\displaystyle \sum xy}=311.292,\text{ and }r\approx -\mathrm{0.976.}$

(b) Use a 1% level of significance to test the claim that $\rho <0$.

(c) Verify that $S_e\approx 0.1660,a\approx 3.366,\text{ and }b\approx -\mathrm{0.0544.}$

(d) Find the predicted optimal time in hours for a dive depth of x = 18 meters.

(e) Find an 80% Confidence interval for y when x = 18 meters.

(f) Use a 1% level of significance to test the claim that $\beta <0$.