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 13.1 23.3 31.2 38.3 51.3 20.5 22.7 y 2.78 1.98 1.78 1.03 0.75 2.38 2.20 (a) Find Σx, Σy, Σx2, Σy2, Σxy, and r. (Round r to three decimal places.) Σx = 200.4 Correct: Your answer is correct. Σy = 12.9 Correct: Your answer is correct. Σx2 = 6722.06 Correct: Your answer is correct. Σy2 = 26.945 Correct: Your answer is correct. Σxy = 314.742 Correct: Your answer is correct. r = -.976 Correct: Your answer is correct. (b) Use a 1% level of significance to test the claim that ρ < 0. (Round your answers to two decimal places.) t = -8.11 Incorrect: Your answer is incorrect. critical t = Conclusion Fail to reject the null hypothesis. There is insufficient evidence that ρ < 0. Reject the null hypothesis. There is sufficient evidence that ρ < 0. Fail to reject the null hypothesis. There is sufficient evidence that ρ < 0. Reject the null hypothesis. There is insufficient evidence that ρ < 0. (c) Find Se, 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 = 36 meters. (Round your answer to two decimal places.) hr (e) Find an 80% confidence interval for y when x = 36 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 insufficient evidence that β < 0. 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. (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 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 within the confidence interval. For a 1 meter increase in depth, the optimal time decreases by an amount that falls outside the confidence interval.

Answer 1

	n=	7.0000
	X̅=ΣX/n	28.6286
	Y̅=ΣY/n	1.8429
	sx=(√(Σx2-(Σx)2/n)/(n-1))=	12.8121
	sy=(√(Σy2-(Σy)2/n)/(n-1))=	0.7271
	Cov=sxy=(ΣXY-(ΣXΣY)/n)/(n-1)=	-9.0944
	r=Cov/(Sx*Sy)=	-0.9762
	slope= β̂1 =r*Sy/Sx=	-0.0554
	intercept= β̂0 ='y̅-β1x̅=	3.4290

b)

t=r*(√(n-2)/(1-r2))=	-10.07
critical t =	-3.36

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

c)

Se =√(SSE/(n-2))=	0.17261
a=	3.42898
b=	-0.05540

d)

predicted val=3.429+36*-0.055=

1.43

e_)

standard error of PI=s*√(1+1/n+(x0-x̅)2/Sxx)=		0.1889
for 80 % CI value of t =	1.476
margin of error E=t*std error   =		0.279
lower confidence bound=xo-E =		1.16
Upper confidence bound=xo+E=		1.71

f)

t=r*(√(n-2)/(1-r2))=	-10.07
critical t =	-3.36

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

g)

std error of slope =se(β1) =s/√Sxx=		0.0055
for 90 % CI value of t =	2.015
margin of error E=t*std error   =		0.011
lower confidence bound=xo-E =		-0.066
Upper confidence bound=xo+E=		-0.044

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