Question

Aviation and high-altitude physiology is a specialty in the study of medicine. Let x = partial pressure of oxygen in the alveoli (air cells in the lungs) when breathing naturally available air. Let y = partial pressure when breathing pure oxygen. The (x, y) data pairs correspond to elevations from 10,000 feet to 30,000 feet in 5000 foot intervals for a random sample of volunteers. Although the medical data were collected using airplanes, they apply equally well to Mt. Everest climbers (summit 29,028 feet). x 7.2 4.8 4.2 3.3 2.1 (units: mm Hg/10) y 43.0 31.5 26.2 16.2 13.9 (units: mm Hg/10)

(a) Verify that Σx = 21.6, Σy = 130.8, Σx2 = 107.82, Σy2 = 3983.34, Σxy = 653.49, and r ≈ 0.980. Σx Σy Σx2 Σy2 Σxy r

(b) Use a 1% level of significance to test the claim that ρ > 0. (Use 2 decimal places.) t critical t

(c) Verify that Se ≈ 2.7422, a ≈ -0.173, and b ≈ 6.096. Se a b

(d) Find the predicted pressure when breathing pure oxygen if the pressure from breathing available air is x = 3.7. (Use 2 decimal places.)

(e) Find a 95% confidence interval for y when x = 3.7. (Use 1 decimal place.) lower limit upper limit

(f) Use a 1% level of significance to test the claim that β > 0. (Use 2 decimal places.) t critical t

(g) Find a 95% confidence interval for β and interpret its meaning. (Use 2 decimal places.) lower limit upper limit

Answer 1

a)

ΣX =	21.600
ΣY=	130.800
ΣX2 =	107.820
ΣY2 =	3983.340
ΣXY =	653.490
r =	0.980

b)

test statistic t =		r*(√(n-2)/(1-r2))=		8.47
t crit =				4.54

c)

Se =√(SSE/(n-2))=				2.74220
a=				-0.173
b=				6.096

d)

predicted value =

22.38

e)

std error of confidence interval =				s*√(1+1/n+(x0-x̅)2/Sxx)=			3.0369
for 95 % confidence and 3degree of freedom critical t=							3.182
lower limit =		12.7
uppr limit =		32.0

f)

test statistic t =		r*(√(n-2)/(1-r2))=		8.47
t crit =				4.54

g)

std error of slope sb1 =				s/√SSx=		0.7199
for 95 % confidence and -2degree of freedom critical t=						3.1820
95% confidence interval =b1 -/+ t*standard error=					(3.805,8.386)
lower limit =		3.81
uppr limit =		8.39