Question

Consider the following data for two variables, x and y.

x	9	32	18	15	26
y	9	19	21	17	22

(a) Develop an estimated regression equation for the data of the form  ŷ = b₀ + b₁x. (Round b₀ to two decimal places and b₁ to three decimal places.)

ŷ =  

Comment on the adequacy of this equation for predicting y. (Use α = 0.05.)

The high p-value and low coefficient of determination indicate that the equation is inadequate

(b) Develop an estimated regression equation for the data of the form ŷ = b₀ + b₁x + b₂x².  (Round b₀ to two decimal places and b₁ to three decimal places and b₂ to four decimal places.)

ŷ =

The low p-value and high coefficient of determination indicate that the equation is adequate.

(c) Use the model from part (b) to predict the value of y when x = 20.  (Round your answer to two decimal places.)

Answer 1

a)

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.712443512
R Square	0.507575758
Adjusted R Square	0.343434343
Standard Error	4.194753818
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	1	54.41212121	54.41212121	3.092307692	0.176903591
Residual	3	52.78787879	17.5959596
Total	4	107.2
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	9.478787879	4.984739491	1.901561335	0.153395335	-6.384877897	25.34245365	-6.384877897	25.34245365
X	0.406060606	0.230913598	1.758495861	0.176903591	-0.32880952	1.140930732	-0.32880952	1.140930732

Y = 9.48 + 0.406 * X

P value = 0.1769 > 0.05, R^2 = 0.5076, It is inadequate

b)

Y	X	X^2
9	9	81
19	32	1024
21	18	324
17	15	225
22	26	676

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.995531446
R Square	0.991082859
Adjusted R Square	0.982165718
Standard Error	0.691345607
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	2	106.2440825	53.12204125	111.1435692	0.008917141
Residual	2	0.955917497	0.477958748
Total	4	107.2
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-11.98263171	2.218603991	-5.400978166	0.032613364	-21.52851423	-2.436749196	-21.52851423	-2.436749196
X	2.856337856	0.238352303	11.98368054	0.006891472	1.830790668	3.881885043	1.830790668	3.881885043
X^2	-0.059107565	0.005675962	-10.41366526	0.009095693	-0.083529257	-0.034685873	-0.083529257	-0.034685873

Y = -11.98 + 2.856 * X - 0.0591 * X^2

P value = 0.0089 < 0.05, R^2 = 0.9911, it is adequate

c)

If X = 20

Y = -11.98 + 2.856 * 20 - 0.0591 * 20^2 = 21.50