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 (data96.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       61.372  54      Large
2       42.7536 33      Small
3       45.7298 57      Small
4       59.9747 63      Small
5       66.9318 190     Large
6       47.0695 112     Small
7       49.306  41      Large
8       41.0451 49      Large
9       55.3925 29      Large
10      51.586  51      Small
11      48.8118 34      Large
12      47.6549 49      Small
13      51.9155 58      Small
14      49.4336 71      Large
15      49.0055 59      Large
16      67.1645 54      Large
17      41.0187 146     Large
18      66.6989 64      Small
19      37.3414 57      Large
20      41.1314 44      Large
21      68.2611 54      Large
22      49.2968 146     Small
23      41.31   71      Large
24      51.4378 131     Small
25      39.8553 92      Large
26      46.3618 137     Small
27      49.451  31      Small
28      41.2582 85      Large
29      56.6397 46      Large
30      43.6494 98      Large
31      43.9767 125     Small
32      44.4695 87      Large
33      45.9359 95      Large
34      42.7246 100     Small
35      45.1822 111     Large
36      67.6785 126     Large
37      47.3643 115     Large
38      44.4952 41      Small
39      41.0943 58      Large
40      43.894  133     Small
41      49.3584 49      Small
42      48.877  86      Small
43      55.4153 37      Large
44      52.4565 25      Small
45      79.1435 85      Large
46      47.3674 121     Small
47      37.8021 81      Large
48      38.0888 31      Large
49      40.4484 146     Small
50      45.4833 15      Large
51      53.525  38      Large
52      52.6821 102     Large
53      42.9239 18      Large
54      40.1348 168     Small
55      75.9808 72      Small
56      37.1931 24      Large
57      48.2541 36      Small
58      49.557  44      Large
59      79.727  28      Small
60      53.8476 56      Large

Answer 1

Here we have given that,

n= no of women who work in Indiana banks= 60

(a)

we want to plot the scatter plot y on x

y is response variable wages

and x is predictor variable LOS

Interpretation : the above scatter plot shows that there is the linear correlation between y and x variables so we can use regression analysis in this case.

(b)

Sum of X = 3003.9396

Sum of Y = 4459

Mean X = 50.0657

Mean Y = 74.3167

Sum of squares (SS_X) = 6129.8763

Sum of products (SP) = -1386.4651

Regression Equation = ŷ = bX + a

b = SP/SS_X = -1386.47/6129.88 = -0.22618

a = M_Y - bM_X = 74.32 - (-0.23*50.07) = 85.6406

ŷ = -0.22618X + 85.6406

Source	DF	Sum of Squares	Mean Square	F Statistic	P-value
Regression (between ŷiand yi)	1	313.5929	313.5929	0.1807	0.6723
Residual (between yiand ŷi)	58	100631.3904	1735.0240
Total(between yiand yi)	59	100944.9833	1710.9319

	Coeff	SE	t-stat	lower t0.025(58)	upper t0.975(58)	Stand Coeff	p-value	VIF
b	85.6406	27.1733	3.1516	31.2474	140.0338	0.000	0.002569
X1	-0.2262	0.5320	-0.4251	-1.2911	0.8388	-0.05574	0.6723	1.0000

T = 3.1516, p-value = 0.002569. Hence b is significantly different from zero.

t = 3.1516

p = 0.002569

(c)

The slope means that for every unit increasing the value of LOS the wages decreases by 0.22618 units

The relationship between wages and length of service is consistent change occur.

(d)

95% confidence interval for the slope:

	Lower 95%	Upper 95%
Intercept	31.2474	140.0338
los	-1.2911	0.8388