Question

3.) The US navy has done a lot of research on the optimal time for a scuba diver to be on the bottom of the ocean. 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 y = optimal time in hours. A random sample of divers gave the following data (based on information taken from 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

Use a 1% level of significance to test the claim that ρ < 0. (I do need you to step through the five steps of hypothesis testing that we have been using in class. You can use SPSS to calculate r and as a check to what you find when you complete the hypothesis test).

Answer 1

Consider the Variable that

X : Depth of dive in meters.

Y: Optimal time in hours.

Let : Population correlation coefficient between X and Y.

We have to test the cliam that

whether or not there is negative correlation between X and Y.

i.e. Null Hypothesis -

against

Alternative hypothesis -

X	Y	X^2	Y^2	XY
14.1	2.58	198.81	6.6564	36.378
24.3	2.08	590.49	4.3264	50.544
30.2	1.58	912.04	2.4964	47.716
38.3	1.03	1466.89	1.0609	39.449
51.3	0.75	2631.69	0.5625	38.475
20.5	2.38	420.25	5.6644	48.79
22.7	2.2	515.29	4.84	49.94
201.4	12.6	6735.46	25.607	311.292

Correlation coefficient of r is

Value of correlation coefficient r = -0.9762.

Alpha: level of significance = 0.01

d.f. = n-2 =7-2=5

From statistical table At 1% and d.f. =5 , critical value of r is 0.875

Since test is left tailed

Value of correlation coefficient r < - Critical value of r.

Decision: We reject the null hypothesis at 1% l.o.s.

Conclusion : There is sufficient evidence support to cliam that there is negative correlation between X and Y.