Question

The following data is a part of the California district test score average and student-teacher-ratio.

Observation Number	TS	STR
1	690.8	17.89
2	661.2	21.52
3	643.6	18.70
4	647.7	17.36
5	640.8	18.67
6	605.6	21.41
7	606.8	19.50
8	609.0	20.89
9	612.5	19.95
10	612.7	20.81
11	615.8	21.24
12	616.3	21.00
13	616.3	20.60
14	616.3	20.01
15	616.5	18.03
16	617.3	20.25
17	618.1	16.98
18	618.3	16.51
19	619.8	22.70
20	620.3	19.91

Using the data given we want to retrieve a linear regression line that reflects the effect of STR on TS

1. Calculate OLS estimator of TS.
2. Construct 95% confidence interval for Test score and Student teacher ratio

Answer 1

a)

Observation Number	TS (y)	STR (x)	xi^2	yi^2	xi*yi
1	690.8	17.89	320.0521	477204.6	12358.41
2	661.2	21.52	463.1104	437185.4	14229.02
3	643.6	18.7	349.69	414221	12035.32
4	647.7	17.36	301.3696	419515.3	11244.07
5	640.8	18.67	348.5689	410624.6	11963.74
6	605.6	21.41	458.3881	366751.4	12965.9
7	606.8	19.5	380.25	368206.2	11832.6
8	609	20.89	436.3921	370881	12722.01
9	612.5	19.95	398.0025	375156.3	12219.38
10	612.7	20.81	433.0561	375401.3	12750.29
11	615.8	21.24	451.1376	379209.6	13079.59
12	616.3	21	441	379825.7	12942.3
13	616.3	20.6	424.36	379825.7	12695.78
14	616.3	20.01	400.4001	379825.7	12332.16
15	616.5	18.03	325.0809	380072.3	11115.5
16	617.3	20.25	410.0625	381059.3	12500.33
17	618.1	16.98	288.3204	382047.6	10495.34
18	618.3	16.51	272.5801	382294.9	10208.13
19	619.8	22.7	515.29	384152	14069.46
20	620.3	19.91	396.4081	384772.1	12350.17
Total	12505.7	393.93	7813.52	7828232	246109.5
Average	625.285	19.6965

Sxx	54.47725
Syy	8605.366
Sxy	-209.029

let the ols b y = b₀ + b₁x

b₁ = Sxy/Sxx = -209.029/54.477 = -3.837

b₀ = = 625.285 - (-3.837*19.697) = 700.86

y = 700.86 - 3.837*x

The ordinary least squares regression line will be

TS = 700.86 - 3.837*STR

b) 95% CI calculated from minitab output

One-Sample T: TS (y), STR (x)

Descriptive Statistics

Sample	N	Mean	StDev	SE Mean	95% CI for μ
TS (y)	20	625.28	21.28	4.76	(615.32, 635.25)
STR (x)	20	19.697	1.693	0.379	(18.904, 20.489)