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 (data390.dat written below) 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 =

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

worker  wages   los     size
1       57.5591 49      Large
2       38.9148 114     Small
3       72.6089 105     Small
4       47.9553 75      Small
5       49.325  74      Large
6       45.0708 35      Small
7       65.2774 20      Large
8       81.4614 164     Large
9       74.4743 24      Large
10      66.7027 24      Small
11      49.7468 118     Large
12      43.3773 104     Small
13      51.8696 24      Small
14      46.9347 79      Large
15      45.1459 41      Large
16      42.0055 72      Large
17      40.3196 73      Large
18      40.5613 53      Small
19      47.2716 61      Large
20      38.8769 16      Large
21      65.738  135     Large
22      49.8769 116     Small
23      39.6822 100     Large
24      55.0231 75      Small
25      39.7514 32      Large
26      62.4771 63      Small
27      43.6881 37      Small
28      68.4972 29      Large
29      66.8898 33      Large
30      40.2356 187     Large
31      43.8013 122     Small
32      51.5445 55      Large
33      50.3812 75      Large
34      60.0526 83      Small
35      44.1453 28      Large
36      87.1545 96      Large
37      59.3941 140     Large
38      37.2679 37      Small
39      46.1983 137     Large
40      41.6133 150     Small
41      40.0572 59      Small
42      73.7666 63      Small
43      40.0042 134     Large
44      38.7618 79      Small
45      39.3117 39      Large
46      43.6889 47      Small
47      64.5969 54      Large
48      65.2548 21      Large
49      38.1001 55      Small
50      66.9966 21      Large
51      64.3793 76      Large
52      39.8683 72      Large
53      49.3329 48      Large
54      38.0247 17      Small
55      54.6429 70      Small
56      47.4064 71      Large
57      50.9659 30      Small
58      40.9046 32      Large
59      58.1326 145     Small
60      37.0958 29      Large

Answer 1

a)

b)

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
49	57.5591	452.98	36.68	-128.90
114	38.9148	1911.15	158.46	-550.30
105	72.6089	1205.25	445.47	732.74
75	47.9553	22.25	12.58	-16.73
74	49.325	13.81	4.74	-8.09
35	45.0708	1244.91	41.37	226.94
20	65.2774	2528.41	189.74	-692.64
164	81.4614	8782.81	897.52	2807.63
24	74.4743	2142.15	527.69	-1063.20
24	66.7027	2142.15	231.04	-703.51
118	49.7468	2276.88	3.08	-83.79
104	43.3773	1136.81	66.02	-273.96
24	51.8696	2142.15	0.13	-16.98
79	46.9347	75.98	20.87	-39.82
41	45.1459	857.51	40.41	186.15
72	42.0055	2.95	90.20	-16.30
73	40.3196	7.38	125.06	-30.38
53	40.5613	298.71	119.71	189.10
61	47.2716	86.18	17.90	39.28
16	38.8769	2946.68	159.41	685.37
135	65.738	4188.25	202.64	921.26
116	49.8769	2090.01	2.64	-74.33
100	39.6822	883.08	139.72	-351.27
75	55.0231	22.247	12.393	16.604
32	39.7514	1465.614	138.094	449.880
63	62.4771	53.047	120.437	-79.930
37	43.6881	1107.780	61.068	260.097
29	68.4972	1704.314	288.812	-701.589
33	66.8898	1390.047	236.762	-573.681
187	40.2356	13622.780	126.948	-1315.061
122	43.8013	2674.614	59.312	-398.292
55	51.5445	233.580	0.002	-0.638
75	50.3812	22.247	1.258	-5.290
83	60.0526	161.714	73.100	108.726
28	44.1453	1787.880	54.132	311.096
96	87.1545	661.347	1271.049	916.845
140	59.3941	4860.414	62.274	550.160
37	37.2679	1107.780	202.630	473.782
137	46.1983	4451.114	28.137	-353.894
150	41.6133	6354.747	97.801	-788.352
59	40.0572	127.3136111	131.0000425	129.1436738
63	73.7666	53.04694444	495.68013	-162.1552229
134	40.0042	4059.813611	132.2160772	-732.6476846
79	38.7618	75.98027778	162.3311699	-111.0583963
39	39.3117	978.6469444	148.6210906	381.3758988
47	43.6889	542.1136111	61.05586113	181.9318921
54	64.5969	265.1469444	171.4574189	-213.2168163
21	65.2548	2428.846944	189.1195668	-677.7480963
55	38.1001	233.5802778	179.6303569	204.8367854
21	66.9966	2428.846944	240.0601625	-763.5898062
76	64.3793	32.68027778	165.8061837	73.61108708
72	39.8683	2.946944444	135.3598451	-19.97242958
48	49.3329	496.5469444	4.708140531	48.35093375
17	38.0247	2839.113611	181.6571579	718.1540988
70	54.6429	0.080277778	9.860699031	-0.88971625
71	47.4064	0.513611111	16.77987851	-2.935699583
30	50.9659	1622.746944	0.288181081	21.62510042
32	40.9046	1465.613611	112.3202535	405.7315521
145	58.1326	5582.580278	43.95524252	495.3621604
29	37.0958	1704.313611	207.559488	594.7658871

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	4217	3090.1635	104060.1833	9156.8	1179.41
mean	70.28	51.50	SSxx	SSyy	SSxy

sample size ,   n =   60          
here, x̅ = Σx / n=   70.28   ,     ȳ = Σy/n =   51.50  
                  
SSxx =    Σ(x-x̅)² =    104060.1833          
SSxy=   Σ(x-x̅)(y-ȳ) =   1179.4          
                  
estimated slope , ß1 = SSxy/SSxx =   1179.4   /   104060.183   =   0.0113
                  
intercept,   ß0 = y̅-ß1* x̄ =   50.7061          
                  
so, regression line is wages =   50.706   +   0.011   *LOS
--

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   60              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    12.556   /√   104060.18   =   0.0389
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    0.0113   /   0.0389   =   0.2912
                  
t-critical value=    2.002   [excel function: =T.INV.2T(α,df) ]          
Degree of freedom ,df = n-2=   58              
p-value =    0.7719             
decison :    p-value>α , do not reject Ho              
Conclusion:   do not Reject Ho and conclude that slope is not significanty different from zero              

d)

confidence interval for slope                  
α=   0.05              
t critical value=   t α/2 =    2.002   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    12.55568   /√   104060.18   =   0.039
                  
margin of error ,E= t*std error =    2.002   *   0.039   =   0.078
estimated slope , ß^ =    0.0113              
                  
                  
lower confidence limit = estimated slope - margin of error =   0.0113   -   0.078   =   -0.0666
upper confidence limit=estimated slope + margin of error =   0.0113   +   0.078   =   0.0892