Question

Listed below are altitudes​ (thousands of​ feet) and outside air temperatures​ (°F) recorded during a flight. Find the​ (a) explained​ variation, (b) unexplained​ variation, and​(c) indicated prediction interval. There is sufficient evidence to support a claim of a linear​ correlation, so it is reasonable to use the regression equation when making predictions. For the prediction​ interval, use a​ 95% confidence level with the altitude of 6327 ft​ (or 6.327 thousand​ feet).

Altitude	4	11	15	20	29	31	33
Temperature	56	30	21	−2	−31	−41	-52

Answer 1

Altitude, X	Temperature, Y	XY	X²	Y²
4	56	224	16	3136
11	30	330	121	900
15	21	315	225	441
20	-2	-40	400	4
29	-31	-899	841	961
31	-41	-1271	961	1681
33	-52	-1716	1089	2704
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
143	-19	-3057	3653	9827

Sample size, n =	7
x̅ = Ʃx/n =	20.42857
y̅ = Ʃy/n =	-2.71429
SSxx = Ʃx² - (Ʃx)²/n =	731.714
SSyy = Ʃy² - (Ʃy)²/n =	9775.43
SSxy = Ʃxy - (Ʃx)(Ʃy)/n =	-2668.86

a) Explained​ variation, SSR = SSxy²/SSxx = 9734.3985

b) Unexplained​ variation, SSE = SSyy - SSxy²/SSxx = 41.0301

c) Slope, b = SSxy/SSxx = -3.64740

y-intercept, a = y̅ -b* x̅ = 71.79695

Regression equation :   

ŷ = 71.797 + (-3.6474) x  

Predicted value of y at x = 6.327

ŷ = 71.797 + (-3.6474) * 6.327 = 48.7198  

Significance level, α = 0.05

Critical value, t_c = T.INV.2T(0.05, 5) = 2.5706  

Standard error, se = √(SSE/(n-2)) = 2.8646

95% prediction​ interval :