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 (data9.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.

(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       51.7094 69      Large
2       71.0128 60      Small
3       70.07   202     Small
4       49.9388 89      Small
5       51.4523 81      Large
6       61.5483 94      Small
7       45.4168 55      Large
8       53.4017 88      Large
9       42.3147 155     Large
10      46.2871 86      Small
11      63.229  112     Large
12      57.062  72      Small
13      45.8663 32      Small
14      42.0388 35      Large
15      43.3518 57      Large
16      54.3362 39      Large
17      62.1635 47      Large
18      42.8431 89      Small
19      68.4515 42      Large
20      44.4342 65      Large
21      43.6074 62      Large
22      40.2586 28      Small
23      58.7744 75      Large
24      51.7969 67      Small
25      73.4367 168     Large
26      46.8493 86      Small
27      49.9769 44      Small
28      44.8422 93      Large
29      44.7397 113     Large
30      51.0961 25      Large
31      76.9333 118     Small
32      49.2112 109     Large
33      49.1286 43      Large
34      56.6601 74      Small
35      59.466  85      Large
36      37.9853 146     Large
37      39.2893 88      Large
38      37.1191 81      Small
39      53.4795 57      Large
40      68.418  88      Small
41      55.6763 45      Small
42      60.8119 73      Small
43      61.1519 113     Large
44      52.1887 47      Small
45      64.3686 33      Large
46      77.7875 188     Small
47      98.2949 75      Large
48      70.8228 81      Large
49      48.0061 70      Small
50      44.3186 22      Large
51      55.4166 59      Large
52      47.1434 58      Large
53      49.7145 78      Large
54      59.1692 57      Small
55      48.7496 45      Small
56      61.6285 71      Large
57      73.1227 26      Small
58      44.0953 65      Large
59      51.2836 30      Small
60      37.4581 55      Large

Answer 1

a) Plot wages versus LOS

get the data into an Excel sheet, as below

select the columns corresponding to LOS,wages and use insert--->scatter

get this raw graph

format as needed

We can see an overall linear trend that the wages increase with the increase in LOS. Hence a linear regression may be appropriate.

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

using data-->data analysis-->regression

get this

ans: The least square regression line is

Summarize the significance test for the slope.

The hypotheses are

The test statistics and P-values are picked from the output

ans:

t=1.9101

P-Value=0.0611

What do you conclude?

We will reject the null hypothesis if the p-value is less than the level of significance

Here, the p-value is 0.0611 and it is greater than the significance level alpha=0.05. Hence we do not reject the null hypothesis.

We conclude that at 5% level of significance there is no sufficient evidence to claim that LOS can explain wages.

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

The value of the slope is +0.0759. The positive sign indicates that the LOS and wages move in the same direction. That is if the length of service (LOS) increases by 1 month, the wages per week would increase by 0.08 (dollars?, as the real wages are multiplies by a constant)

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

The confidence interval is provided in the output is [-0.0036, 0.1555]

The following is the calculation

95% confidence interval corresponds to the significance level

The right tail critical value is .

Number of observations is 60. The degrees of freedom for t is 60-2=58

Using t table for df=58 (pick the closest value df=60) and area under the right tail=0.025 (total area under 2 tails=0.05) we get

the estimate of the slope is

the standard error of slope is

The 95% confidence interval is