Question

Reserve Problems Chapter 11 Section 2 Problem 2 The peanut crop was harvested from five fields of various area. The following data are the mass of the crop from each field y (in kilograms) and the field area x (in hectares). y 7450 15650 13480 1984012820 x 2.44 4.09 3.47 4.59 2.65 Round your intermediate answers to four decimal places (e.g. 98.7654). (a) Fit the simple linear regression model using the method of least squares. Find the estimate of o?. Round your answer to the nearest integer (e.g. 9876). (b) What change in the mean mass is expected when the field area changes by 1 hectare? Round your answer to the nearest integer (e.g. 9876). B (c) Calculate the fitted value of y corresponding to x = 4.59. Find the corresponding residual. Round your answer to the nearest integer (e.g. 9876). Round your answer to the nearest integer (e.g. 9876). e = (d) Estimate the mean mass of the crop harvested from 4.0 hectares. Round your answer to the nearest integer (e.g. 9876).

Answer 1

	x	y	X-X	y-ý	(y-7) (x - 1)	(x - 2)2	(y – 5)2
	2.44	7450	-1.008	-6398	6449.184	1.016	40934404
	4.09	15650	0.642	1802	1156.884	0.412	3247204
	3.47	13480	0.022	-368	-8.096	0.000	135424
	4.59	19840	1.142	5992	6842.864	1.304	35904064
	2.65	12820	-0.798	-1028	820.344	0.637	1056784
Total	17.24	69240	0	0	15261.18	3.370	81277880
Mean	3.448	13848

The regression equation between two variables can be given by following equation

ỹ = 60 + b1 * *

The estimate of b0 and b1 can be obtained by least square method.

The least square estimate of b1 can be given by following expression

b1 : Σy - y) * (x – x) Σ(x – x)2

b1 15261.18 3.370

b1 = 4528.97

The estimate of b0 can be given by

b0 = 7 – b1*

b0 = 13848 - (4528.97) * 3.448

b0 = -1767.89

The final estimated simple linear regression model can be given as

ŷ = -1767.89+ 4528.97 * x

The ANOVA table of fitted regression is given by

Source	Df	SS	MS	F	p-value
Regression	1	b1*Sxy=4529*15261.18          = 69117428	69117428	17.05	0.0258
Error	3	12160452	4053484
Total	n-1=5-1=4	SYY=81277880

Answer(a):

The estimate of σ²can be obtained by MSE.

Here we have, MSE =4053484

Hence

2 = 4053484

Answer(b):

The expected change in the mean mass when the field are changes by 1 hectare is given by the slope of simple linear regression model.

Hence

B1 = 4529

Answer(c):

The fitted value of y corresponding to x=4.59 can be obtained as

ŷ = -1767.89 +4528.97 * 4.59

y = 19020

The residual can be obtained as

e = y - ☺

e = 19840 – 19020

e = 820

Answer(d):

The estimate of mean mass of the crop harvested from 4 hectares can be obtained as

Y = -1767.89+ 4528.97 * 4

y = 16348