Question

Consider the simple linear regression model y = 10+25x+e where the random error term is normally and independently distributed with mean zero and standard deviation 2. Do NOT use software, generate a sample of eight observations, one each at the levels x = 10, 12, 14, 16, 18, 20, 22, and 24.

DO NOT USE SOFTWARE!

A.Fit the linear regression model by least squares and find the

estimates of the slope and intercept.

B.Find the estimate of s^2.

C. Find the standard errors of the slope and intercept.

D. do NOT use software, generate a sample of 16 observations, two each at the same levels of x used previously. Fit the model using least squares.

E. Find the estimate of s2 for the new model in part (d). Compare this to the estimate obtained in part (b). What impact has the increase in sample size had on the estimate?

F. Find the standard errors of the slope and intercept using the new model from part (d). Compare these standard errors to the ones that you found in part (c). What impact has the increase in sample size had on the estimated standard errors?

Answer 1

a) The least squares estimates of the intercept and slope in the simple linear regression model are

Using the data point x = 10, 12, 14, 16, 18, 20, 22, and 24. and equation y = 10+25x+e

e is normally distributed with mean 0 and standard deviation of 2. Since we need to do it manually without software. I have picked random values ranging from -6 to 6.

	X	e	Y	X^2	Y^2	XY
1	10	2.820463	262.8205	100	69074.6	2628.205
2	12	0.666063	310.6661	144	96513.4	3727.993
3	14	-0.10547	359.8945	196	129524.1	5038.523
4	16	-0.40763	409.5924	256	167765.9	6553.478
5	18	-5.68276	454.3172	324	206404.2	8177.71
6	20	-1.26842	508.7316	400	258807.8	10174.63
7	22	-3.15955	556.8404	484	310071.3	12250.49
8	24	0.943738	610.9437	576	373252.3	14662.65
mean	17	-0.7742	434.2258	310	201426.7	7901.71
Sum	136	-6.19357	3473.806	2480	1611413	63213.68

the predicted least square model is  

B. Find the estimate of s^2.

The formula for calculating error sum of square and estimate of s^2 is as follows

using the predicted least square model is  , we calculated respective y values

Count	Y	y hat	e	e^2
1	262.8205	260.9354	1.885111	3.553644
2	310.6661	310.4469	0.219154	0.048028
3	359.8945	359.9585	-0.06394	0.004088
4	409.5924	409.47	0.12235	0.014969
5	454.3172	458.9816	-4.66434	21.75605
6	508.7316	508.4931	0.238436	0.056852
7	556.8404	558.0047	-1.16425	1.355485
8	610.9437	607.5163	3.427481	11.74763
			SUM	38.53674

C. Find the standard errors of the slope and intercept.

In simple linear regression, the estimated standard error of the slope and the estimated standard error of the intercept are

	X	x - x mean	(x - x mean)^2	X^2
1	10	-7	49	100
2	12	-5	25	144
3	14	-3	9	196
4	16	-1	1	256
5	18	1	1	324
6	20	3	9	400
7	22	5	25	484
8	24	7	49	576
mean	17	0	21	310
sum	136	0	168	2480

d) generate a sample of 16 observations, two each at the same levels of x used previously. Fit the model using least squares.

	X	e	Y	X^2	Y^2	XY
1	10	5.175403	265.1754	100	70317.99	2651.754
2	10	-0.56329	259.4367	100	67307.41	2594.367
3	12	-0.80382	309.1962	144	95602.28	3710.354
4	12	-0.75815	309.2418	144	95630.52	3710.902
5	14	-0.95333	359.0467	196	128914.5	5026.653
6	14	1.87862	361.8786	196	130956.1	5066.301
7	16	1.949255	411.9493	256	169702.2	6591.188
8	16	1.142205	411.1422	256	169037.9	6578.275
9	18	-1.26444	458.7356	324	210438.3	8257.24
10	18	1.37192	461.3719	324	212864	8304.695
11	20	-0.19699	509.803	400	259899.1	10196.06
12	20	1.891849	511.8918	400	262033.3	10237.84
13	22	-0.19273	559.8073	484	313384.2	12315.76
14	22	-4.55689	555.4431	484	308517	12219.75
15	24	-0.15097	609.849	576	371915.8	14636.38
16	24	3.524011	613.524	576	376411.7	14724.58
mean	16.53333	0.264576	423.5979	292.2667	191101.4	7473.167
Sum	248	3.968634	6353.969	4384	2866521	112097.5

y = 11.74769 + 24.92474