Question

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.

x	14.1	22.3	31.2	38.3	51.3	20.5	22.7
y	2.48	1.98	1.48	1.03	0.75	2.38	2.20

(a) Find Σx, Σy, Σx², Σy², Σxy, and r. (Round r to three decimal places.)

Σx	=
Σy	=
Σx2	=
Σy2	=
Σxy	=
r	=

(b) Use a 1% level of significance to test the claim that ρ < 0. (Round your answers to two decimal places.)

t	=
critical t	=

Conclusion

Reject the null hypothesis. There is sufficient evidence that ρ < 0.Fail to reject the null hypothesis. There is sufficient evidence that ρ < 0.     Fail to reject the null hypothesis. There is insufficient evidence that ρ < 0.Reject the null hypothesis. There is insufficient evidence that ρ < 0.

(c) Find S_e, a, and b. (Round your answers to five decimal places.)

Se	=
a	=
b	=

(d) Find the predicted optimal time in hours for a dive depth of x = 38 meters. (Round your answer to two decimal places.)
hr

(e) Find an 80% confidence interval for y when x = 38 meters. (Round your answers to two decimal places.)

lower limit	hr
upper limit	hr

(f) Use a 1% level of significance to test the claim that β < 0. (Round your answers to two decimal places.)

t	=
critical t	=

Conclusion

Reject the null hypothesis. There is sufficient evidence that β < 0.Fail to reject the null hypothesis. There is sufficient evidence that β < 0.     Fail to reject the null hypothesis. There is insufficient evidence that β < 0.Reject the null hypothesis. There is insufficient evidence that β < 0.

(g) Find a 90% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)

lower limit
upper limit

Interpretation

For a 1 meter increase in depth, the optimal time increases by an amount that falls within the confidence interval.For a 1 meter increase in depth, the optimal time decreases by an amount that falls outside the confidence interval.     For a 1 meter increase in depth, the optimal time decreases by an amount that falls within the confidence interval.For a 1 meter increase in depth, the optimal time increases by an amount that falls outside the confidence interval.

Answer 1

Answer a)

Answer b)

Formula for Test statistics t = r*sqrt((n-2)/(1-r²))

Test statistics t = -0.969*sqrt((7-2)/(1-0.969²))

Test statistics t = -8.770

Critical value t_c = -3.365 (Value of t corresponding to df =7-2 = 5, and α = 0.01 Left Tail)

H0​: ρ=0

HA​: ρ<0

Now, in this case t < t_c so we reject null hypothesis.

Reject the null hypothesis. There is sufficient evidence that ρ < 0

Answer c)

Explanation

Value of regression coefficient a and b can be calculated as follows:

Formula of Standard Error as given below:

(Y-Y')^2 is calculated as follows:

Standard Error = SQRT(77.45199/(7-2))

Standard Error Se = 3.80663

Answer d)

To obtain predicted optimal time in hours for a dive depth of x = 38 meters, we can substitute x = 38 in regression equation

y = 3.244 - 0.052*38

y = 1.27 hr

We can answer 4 questions only.