M12 Q9

What is the optimal time for a scuba diver to be on the bottomof the ocean? That depends on the depth of the dive. The U.S. Navyhas done a lot of research on this topic. The Navy defines the"optimal time" to be the time at each depth for the best balancebetween length of work period and decompression time aftersurfacing. Let x = depth of dive in meters, and lety = optimal time in hours. A random sample of divers gavethe following data.

x	16.1	26.3	31.2	38.3	51.3	20.5	22.7
y	2.68	2.18	1.48	1.03	0.75	2.38	2.20

(a) Find ?x, ?y, ?x²,?y², ?xy, and r. (Roundr to three decimal places.)

?x	=
?y	=
?x2	=
?y2	=
?xy	=
r	=

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

t	=
critical t	=

Conclusion

Fail to reject the null hypothesis. There is insufficientevidence that ? < 0.

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

Fail to reject the null hypothesis. There is sufficient evidencethat ? < 0.

Reject the null hypothesis. There is insufficient evidence that? < 0.

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

Se	=
a	=
b	=

(d) Find the predicted optimal time in hours for a dive depth ofx = 22 meters. (Round your answer to two decimalplaces.)
hr

(e) Find an 80% confidence interval for y when x= 22 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? < 0. (Round your answers to two decimal places.)

t	=
critical t	=

Conclusion

Fail to reject the null hypothesis. There is insufficientevidence that ? < 0.

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

Reject the null hypothesis. There is insufficient evidence that? < 0.

Fail to reject the null hypothesis. There is sufficient evidencethat ? < 0.

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

lower limit
upper limit

Interpretation

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

For a 1 meter increase in depth, the optimal time decreases byan amount that falls outside the confidenceinterval.     

For a 1 meter increase in depth, the optimal time decreases byan amount that falls within the confidence interval.

For a 1 meter increase in depth, the optimal time increases byan amount that falls outside the confidence interval.