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	23.3	30.2	38.3	51.3	20.5	22.7
y	2.68	2.08	1.48	1.03	0.75	2.38	2.20

(a) Find  r. (Round r to three decimal places.)

(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 four decimal places.)

Se =
a =
b =

(d) Find the predicted optimal time in hours for a dive depth of x = 38 meters. (Round your answer to two decimal places.)
hr

(e) Find an 80% confidence interval for y when x = 38 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 =

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	198.40	12.60	1012.24	3.15	-54.88
mean	28.34	1.80	SSxx	SSyy	SSxy

a)  correlation coefficient ,    r = SSxy/√(SSx.SSy) =   -0.972

b)

t-test statistic = r*√(n-2)/√(1-r²) =        -9.30
DF=n-2 =   5  

critical t-value =    -3.36

c)

std error ,Se =    √(SSE/(n-2)) =    0.1854
      
a = intercept,   ß0 = y̅-ß1* x̄ =   3.3366

b = estimated slope , ß1 = SSxy/SSxx =   -54.9   /   1012.237   =   -0.0542

d)

Predicted Y at X=   38   is          
Ŷ =   3.3366   +   -0.0542   *38=   1.28

e)

standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    0.090              
margin of error,E=t*Std error=t* S(ŷ) =   1.4759   *   0.090   =   0.1326
                  
Confidence Lower Limit=Ŷ +E =    1.276   -   0.133   =   1.14
Confidence Upper Limit=Ŷ +E =   1.276   +   0.133   =   1.41

f)

estimated std error of slope =Se(ß1) = Se/√Sxx =    0.185   /√   1012.24   =   0.0058
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    -0.0542   /   0.0058   =   -9.30
                  
t-critical value=    -3.36 [Excel function: =T.INV(α,df) ]