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 23.3 29.2 38.3 51.3 20.5 22.7 y 2.78 2.18 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 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 = 19 meters. (Round your answer to two decimal places.) hr

(e) Find an 80% confidence interval for y when x = 19 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

a)

Computational Table:

Sample correlation coefficient (r):

r = -0.973

b)

Hypothesis:

   ...... (No linear correlation)

.......Negative correlation

Test statistic:

Degrees of Freedom = n-2 = 7-2 = 5

Critical value:

          .........Using t table

Conclusion:

Test statistic < Critical value, i.e. -9.44 < -3.36, That is Reject Ho at 1% level of significance.

C)

Calculation:

For Slope:

b = -0.05900

For Intercept:

a = 1.86 - (-0.05900)*28.77

a = 3.55476

Therefore, the least square regression line would be,

= 3.55476 - 0.05900

Standard error (Se):

d)

The least square regression line would be,

= 3.55476 - 0.05900 (X)

For X = 19

= 3.55476 - 0.05900 * 19

= 2.43