Question

9.)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	15.1	24.3	32.2	38.3	51.3	20.5	22.7
y	2.78	2.28	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	=
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

Fail to reject the null hypothesis. There is sufficient evidence that ρ < 0.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 = 37 meters. (Round your answer to two decimal places.)
hr

(e) Find an 80% confidence interval for y when x = 37 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.     Reject the null hypothesis. There is sufficient evidence that β < 0.Fail to 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 increases 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 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.

Answer 1

a)

x	y	x2	y2	xy
15.1	2.78	228.01	7.73	41.978
24.3	2.28	590.49	5.20	55.404
32.2	1.78	1036.84	3.17	57.316
38.3	1.03	1466.89	1.06	39.449
51.3	0.75	2631.69	0.56	38.475
20.5	2.38	420.25	5.66	48.79
22.7	2.20	515.29	4.84	49.94

r =

=

=

=

= -0.976453

b) Convert the sample correlation coefficient r = -0.976453 to a t-value

=

=

= -10.121

Now,

H₀:

H₁:

Using the t-distribution table the p-value corresponding to the sample t-value of -10.121 is less than 0.0005.

Since the p-value is less than 0.01, we reject H₀ and conclude there is a positive correlation between x and y.

c) The formulas for Se, a and b are:

=

=

= -0.05873

where and

Here and

a = (1.886) - (-0.05873)(29.2)

= 3.6006

=

= 4.364

Hence S_e = 4.364, a = 3.6006, b = -0.05873

=