Question

We assume that our wages will increase as we gain experience and become more valuable to our employers. Wages also increase because of inflation. By examining a sample of employees at a given point in time, we can look at part of the picture. How does length of service (LOS) relate to wages? The data here (data128.dat) is the LOS in months and wages for 60 women who work in Indiana banks. Wages are yearly total income divided by the number of weeks worked. We have multiplied wages by a constant for reasons of confidentiality.

(a) Plot wages versus LOS. Consider the relationship and whether or not linear regression might be appropriate. (Do this on paper. Your instructor may ask you to turn in this graph.)

(b) Find the least-squares line. Summarize the significance test for the slope. What do you conclude?

Wages =	+  LOS
t =
P =

(c) State carefully what the slope tells you about the relationship between wages and length of service.


(d) Give a 95% confidence interval for the slope.
(  ,  )

worker  wages   los     size
1       55.2975 117     Large
2       55.8644 73      Small
3       42.4397 34      Small
4       38.7346 47      Small
5       52.4622 19      Large
6       48.5854 35      Small
7       38.6446 22      Large
8       41.3771 156     Large
9       51.3591 16      Large
10      42.4237 25      Small
11      37.5306 22      Large
12      43.9234 48      Small
13      63.0731 170     Small
14      48.4594 15      Large
15      49.2384 65      Large
16      62.3888 46      Large
17      40.8046 91      Large
18      41.7538 38      Small
19      49.46   53      Large
20      44.6369 29      Large
21      62.317  49      Large
22      54.5189 21      Small
23      47.2384 112     Large
24      55.1389 71      Small
25      59.6456 63      Large
26      50.3852 25      Small
27      40.6464 105     Small
28      51.6111 48      Large
29      51.1047 128     Large
30      65.7949 72      Large
31      75.6899 77      Small
32      61.2864 34      Large
33      47.7661 100     Large
34      63.3778 18      Small
35      43.8794 24      Large
36      63.9634 164     Large
37      37.3139 77      Large
38      38.8793 100     Small
39      51.4926 53      Large
40      37.3424 59      Small
41      52.8835 92      Small
42      53.1512 32      Small
43      50.1548 45      Large
44      51.355  21      Small
45      74.2463 79      Large
46      57.5963 40      Small
47      50.4146 81      Large
48      58.5332 36      Large
49      52.7574 69      Small
50      66.6319 36      Large
51      64.2785 103     Large
52      49.7572 101     Large
53      53.3475 140     Large
54      63.6077 112     Small
55      41.4925 66      Small
56      46.5037 32      Large
57      59.0737 30      Small
58      52.336  67      Large
59      48.9027 23      Small
60      42.8794 33      Large

Answer 1

a)

The scatter plot between wages and LOS reveals that there may be a very weak relationship between the variables which can be identified by performing least square regression analysis.

b)

The excel output of the regression analysis

SUMMARY OUTPUT

Regression Statistics
Multiple R	0.158061
R Square	0.024983
Adjusted R Square	0.008173
Standard Error	9.165921
Observations	60

ANOVA
	df	SS	MS	F	Significance F
Regression	1	124.8581	124.8581	1.486156	0.227749
Residual	58	4872.819	84.01412
Total	59	4997.677

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	49.31953	2.233785	22.0789	6.53E-30	44.84812	53.79093
los	0.036866	0.030241	1.21908	0.227749	-0.02367	0.097401

Least square regression line is

wages = 0.037 * LOS + 49.319

Significance test for slope

H₀ : Slope is equal to zero.

H₁ : Slope is not equal to zero.

test statistic t = 1.129

p-value = 0.2277

Since p-value is greater than alpha 0.05 we fail to reject null hypothesis and conclude that there is no significant evidence to conclude that slope is different from zero. Which means there doesn't exist linear relationship between the wages and LOS.

c) Interpretation of slope:-

If the length of stay increases by 1 month then the wage of women increases by 0.0369 units.

d) 95% confidence interval for the slope = ( -0.02367, 0.097401)