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	12.1	26.3	32.2	38.3	51.3	20.5	22.7
y	2.78	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.     Reject the null hypothesis. There is insufficient evidence that ρ < 0. Fail to 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 = 32 meters. (Round your answer to two decimal places.)
hr

(e) Find an 80% confidence interval for y when x = 32 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. Fail to reject the null hypothesis. There is insufficient evidence that β < 0.     Fail to reject the null hypothesis. There is sufficient evidence that β < 0. Reject the null hypothesis. There is sufficient 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 decreases 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 increases 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

a)

ΣX =	203.400
ΣY=	12.600
ΣX2 =	6909.060
ΣY2 =	25.967
ΣXY =	310.022
r =	-0.979

b)

test statistic t =		r*(√(n-2)/(1-r2))=		-10.75
t crit =				-3.36

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

c)

Se =√(SSE/(n-2))=				0.16510
a=				3.43195
b=				-0.05616

d)

predicted value =

1.63

e)

std error of confidence interval =				s*√(1+1/n+(x0-x̅)2/Sxx)=			0.1772
for 80 % confidence and 5degree of freedom critical t=							1.476
lower limit =		1.37
uppr limit =		1.89

f)

test statistic t =		r*(√(n-2)/(1-r2))=		-10.75
t crit =				-3.36

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

g)

std error of slope sb1 =				s/√SSx=		0.0052
for 90 % confidence and -2degree of freedom critical t=						2.0150
90% confidence interval =b1 -/+ t*standard error=					(-0.067,-0.046)
lower limit =		-0.067
uppr limit =		-0.046

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