Question

SALARY	EDUC	EXPER	TIME
39000	12	0	1
40200	10	44	7
42900	12	5	30
43800	8	6	7
43800	8	8	6
43800	12	0	7
43800	12	0	10
43800	12	5	6
44400	15	75	2
45000	8	52	3
45000	12	8	19
46200	12	52	3
48000	8	70	20
48000	12	6	23
48000	12	11	12
48000	12	11	17
48000	12	63	22
48000	12	144	24
48000	12	163	12
48000	12	228	26
48000	12	381	1
48000	16	214	15
49800	8	318	25
51000	8	96	33
51000	12	36	15
51000	12	59	14
51000	15	115	1
51000	15	165	4
51000	16	123	12
51600	12	18	12
52200	8	102	29
52200	12	127	29
52800	8	90	11
52800	8	190	1
52800	12	107	11
54000	8	173	34
54000	8	228	33
54000	12	26	11
54000	12	36	33
54000	12	38	22
54000	12	82	29
54000	12	169	27
54000	12	244	1
54000	15	24	13
54000	15	49	27
54000	15	51	21
54000	15	122	33
55200	12	97	17
55200	12	196	32
55800	12	133	30
56400	12	55	9
57000	12	90	23
57000	12	117	25
57000	15	51	17
57000	15	61	11
57000	15	241	34
60000	12	121	30
60000	15	79	13
61200	12	209	21
63000	12	87	33
63000	15	231	15
46200	12	12	22
50400	15	14	3
51000	12	180	15
51000	12	315	2
52200	12	29	14
54000	12	7	21
54000	12	38	11
54000	12	113	3
54000	15	18	8
54000	15	359	11
57000	15	36	5
60000	8	320	21
60000	12	24	2
60000	12	32	17
60000	12	49	8
60000	12	56	33
60000	12	252	11
60000	12	272	19
60000	15	25	13
60000	15	36	32
60000	15	56	12
60000	15	64	33
60000	15	108	16
60000	16	46	3
63000	15	72	17
66000	15	64	16
66000	15	84	33
66000	15	216	16
68400	15	42	7
69000	12	175	10
69000	15	132	24
81000	16	55

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.41198516
R Square	0.16973178
Adjusted R Square	0.16060795
Standard Error	6501.12045
Observations	93
ANOVA
	df	SS	MS	F	Significance F
Regression	1	786253429	786253429	18.60313	4.08E-05
Residual	91	3.85E+09	42264567.1
Total	92	4.63E+09
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	38185.5979	3774.3766	10.117061	1.45E-16	30688.26252	45682.93	30688.26	45682.93
X Variable 1	1280.85932	296.96712	4.31313512	4.08E-05	690.9706164	1870.748	690.9706	1870.748

This data set was obtained by collecting information on a randomly selected sample of 93 employees working at a bank.

SALARY- starting annual salary at the time of hire

EDUC   - number of years of schooling at the time of the hire

EXPER - number of months of previous work experience at the time of hire

TIME    - number of months that the employee has been working at the bank until now

2. Use the least squares method to fit a simple linear model that relates the salary (dependent variable) to education (independent variable).

a- What is your model? State the hypothesis that is to be tested, the decision rule, the test statistic, and your decision, using a level of significance of 5%.

b – What percentage of the variation in salary has been explained by the regression?

c – Provide a 95% confidence interval estimate for the true slope value.

d - Based on your model, what is the expected salary of a new hire with 12 years of education?

e – What is the 95% prediction interval for the salary of a new hire with 12 years of education? Use the fact that the distance value = 0.011286

Answer 1

a. The assumed underlying model is a simple linear model

,  

Looking under the Coefficients column of the fitted model:
are the estimated coefficients

So the fitted model is


Hypothesis testing for the Intercept:



The test statistic, Under the Null hypothesis is t-distributed with n-1 = 93-1 = 92 degrees of freedom

  

Here

Plugging in,
Hence the The obtained test-statistic
At level of significance (5%) i.e.
The critical test-statistc value =

Since the obtained test statistic t = 10.11 is greater than the critical test statistic = 1.986
We reject the NULL hypothesis,
The Intercept term is SIGNIFICANT at 5% level of significance

Similiarly for the slope term
Hypothesis testing for the Slope:



The test statistic, Under the Null hypothesis is t-distributed with n-1 = 93-1 = 92 degrees of freedom

  

Here

Plugging in,
Hence the The obtained test-statistic
At level of significance (5%) i.e.
The critical test-statistc value =

Since the obtained test statistic t = 4.313 is greater than the critical test statistic = 1.986
We reject the NULL hypothesis,
The slope term is SIGNIFICANT at 5% level of significance

b) The percentage of variation of salary explained by the regression is =

The value of for the regression =  

This value is also mentioned directly in the model output as R Square

So the percentage of variation of salary explained by the regression is =

c) 95% confidence interval for the true slope value

The   confidence interval is given by

This value is also directly mentioned in the model output Under Lower 95% and Upper 95%
for X Variable 1

d) Expected salary of a new hire with 12 years of experience =

Putting EDUC = 12 in the fitted regression model, we get

e)  95% prediction interval for the salary of a new hire with 12 years of education

Prediction Interval = Estimated value ± T_{\alpha/2, n-1} * Prediction Error

Prediction Error = Standard Error of the Regression * SQRT(1 + distance value)

Standard Error of Regression

So Prediction Error =

And the Prediction Interval =

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