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	16.1	25.3	28.2	38.3	51.3	20.5	22.7
y	2.68	2.28	1.68	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	=

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

(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 = 22 meters. (Round your answer to two decimal places.)
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 β < 0. (Round your answers to two decimal places.)

t	=
critical t	=

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

lower limit
upper limit

Answer 1

Answer a)

Σx	= 202.4
Σy	= 13
Σx2	= 6728.66
Σy2	= 27.331
Σxy	= 324.862
r	= -0.965

Answer b)

The sample size is n = 7, so then the number of degrees of freedom is df = n-2 = 7 - 2 = 5

The corresponding t-statistic to test for the significance of the correlation is:

The critical t value corresponding to α = 0.01 and df = 5 for a left tailed test is 3.365 (Obtained using t distribution table. Screenshot attached)

t	= 8.27
critical t	= 3.36

Answer c)

a and b has been calculated using values of Σx, Σy, Σx², Σy², Σxy computed in Part a)

a = 3.54051

b = -0.05822

S_e has also been calculated using values of Σx, Σy, Σx², Σy², Σxy computed in Part a)

S_e = 0.20861

Se	= 0.20861
a	= 3.54051
b	= -0.05822

Answer d)

Regression written as:

y = 3.54051 -0.05822*x

At x = 22, y can be derived using regression equation:

y = 3.54051 -0.05822*22

y = 2.26 hr

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