Question

1. 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

The calculation for correraltion coefficient is as below

r= -51.228/(940.894*2.927)= -0.9762

The following needs to be tested:

H0​:ρ=0

HA​:ρ<0

where ρ corresponds to the population correlation.

The sample size is n=7, so then the number of degrees of freedom is

df = n−2=7−2=5

The corresponding critical correlation value rc​ for a significance level of α=0.01, for a left-tailed test is:

rc​=−0.833 (from t-table at 0.01 significance level and df=5)

Observe that in this case, the null hypothesis H0​:ρ=0 is rejected if r<rc​=−0.833.

Based on the sample correlation provided, we have that r=−0.976<rc​=−0.833, from which is concluded that the null hypothesis is rejected. Hence we have enough evidence to calim that the population correlation coefficient is less than 0 (i.e ρ<0).