Question

a. Estimate the regression equation and determine the predicted values of y

b. Use the predicted values and the actual values of y to calculate the residuals

c. Plot the residuals (vertical axis) against the predicted values (horizontal axis).

d. Does it appear that heteroscedasticity is a problem? Explain

e. Draw a histogram of the residuals. Does it appear that the errors are normally distributed? Explain.

f. Use the residuals to identify possible outlier(s)

Weight	Price
27	12.02
28.5	12.04
30.8	12.32
31.3	12.27
31.9	12.49
34.5	12.7
34.7	13
37	13
41	13.17
41	13.19
38.8	13.22
39.3	13.27
34	12.8

please do not use minitab or any statistical software

Answer 1

a. Estimate the regression equation and determine the predicted values of y:

We are given:

Obs.	WeightWeight	PricePrice	X_i^2Xi2​	Y_i^2Yi2​	X_i \cdot Y_iXi​⋅Yi​
1	27	12.02	729	144.4804	324.54
2	28.5	12.04	812.25	144.9616	343.14
3	30.8	12.32	948.64	151.7824	379.456
4	31.3	12.27	979.69	150.5529	384.051
5	31.9	12.49	1017.61	156.0001	398.431
6	34.5	12.7	1190.25	161.29	438.15
7	34.7	13	1204.09	169	451.1
8	37	13	1369	169	481
9	41	13.17	1681	173.4489	539.97
10	41	13.19	1681	173.9761	540.79
11	38.8	13.22	1505.44	174.7684	512.936
12	39.3	13.27	1544.49	176.0929	521.511
13	34	12.8	1156	163.84	435.2
Sum =	449.8	165.49	15818.46	2109.1937	5750.275

=1/n∑​Xi​=449.8/13​=34.6

=34.6

=1/n∑Yi​=165.49​/13=12.73

=12.73

SS_XX​​=∑Xi²​−1/n​(∑Xi​)²=15818.46−1/13​(449.8)²=255.38​

SS_XX=255.38

SS_YY​​=∑​Yi2​−1/n​(∑​Yi​)²=2109.1937−1​/13(165.49)²=2.506​

SS_YY=2.506

SS_XY​​=∑​Xi​Yi​−1/n​(∑​Xi​)(∑​Yi​)=5750.275−1/13​(449.8)(165.49)=24.321​

SS_XY​​=24.321

​​=​SS_XY/SS_XX​​=24.321/255.38=​0.0952​

=0.0952

​​==12.73−0.0952×34.6=9.4349​

=9.4349

the regression equation is:

PREDICTED VALUES:

Obs.	Weight	Price	Predicted Values
1	27	12.02	9.4349+0.0952×27=12.006
2	28.5	12.04	9.4349+0.0952×28.5=12.149
3	30.8	12.32	9.4349+0.0952×30.8=12.368
4	31.3	12.27	9.4349+0.0952×31.3=12.416
5	31.9	12.49	9.4349+0.0952×31.9=12.473
6	34.5	12.7	9.4349+0.0952×34.5=12.72
7	34.7	13	9.4349+0.0952×34.7=12.74
8	37	13	9.4349+0.0952×37=12.959
9	41	13.17	9.4349+0.0952×41=13.34
10	41	13.19	9.4349+0.0952×41=13.34
11	38.8	13.22	9.4349+0.0952×38.8=13.13
12	39.3	13.27	9.4349+0.0952×39.3=13.178
13	34	12.8	9.4349+0.0952×34=12.673

b. Use the predicted values and the actual values of y to calculate the residuals:

Price(observed value)	Predicted Values	Residuals=observed value-predicted value
12.02	12.006	0.014
12.04	12.149	-0.109
12.32	12.368	-0.048
12.27	12.416	-0.146
12.49	12.473	0.017
12.7	12.72	-0.02
13	12.74	0.26
13	12.959	0.041
13.17	13.34	-0.17
13.19	13.34	-0.15
13.22	13.13	0.09
13.27	13.178	0.092
12.8	12.673	0.127

c. Plot the residuals (vertical axis) against the predicted values (horizontal axis):

d. Does it appear that heteroscedasticity is a problem? Explain:

Yes,heteroscedasticity is a problem Heteroscedasticity means unequal scatter.From the Scatter-plot we observe that the variance of the residuals is lower for lower predicted values and higher for higher predicted values.which shows a highly unequal scatter. hence, yes heteroscedasticity is a problem.

e. Draw a histogram of the residuals. Does it appear that the errors are normally distributed? Explain.

Frequency Table
Class	Count
-0.2--0.071	4
-0.07-0.059	5
0.06-0.189	3
0.19-0.319	1

The normal distribution curve is symmetric that is but since the above histrogram is positively skewed hence,the errors are not normally distributed.

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