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Element of economics - Essay Example

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Summary
An insurance manager believes that the number of new policies written annually by his agents is related to the number of years of selling experience these agents have. A random sample of 12 agents revealed the following data:
d) If r = 0, i.e., the coefficient of correlation is…
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Element of economics
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Question: An insurance manager believes that the number of new policies written annually by his agents is related to the number of years of selling experience these agents have. A random sample of 12 agents revealed the following data:
No. of years of experience
No. of new policies written annually
2
17
5
22
7
34
6
37
12
50
9
41
5
13
20
48
13
39
4
20
10
35
20
63
a) Find the least-squares regression line of number of new policies on number of years of experience.
b) Find the coefficient of determination for the line in (a) and comment.
c) Estimate the number of new policies written by an agent with 15 years of experience.
d) What can be said about the slope of a least-squares regression line if the two variables involved have a correlation coefficient of r=0?
e) In the exercise 9.1, can we reliably estimate the number of policies written annually by an agent with 1 year experience? Explain.
f) Interpret the slope of the least-squares regression line in exercise 9.1.
ANSWER:
No. of yrs. of Experience (x)
No. of new policies (y)
x2
y2
xiyi
2
17
4
289
34
5
22
25
484
110
7
34
49
1156
238
6
37
36
1369
222
12
50
144
2500
600
9
41
81
1681
369
5
13
25
169
65
20
48
400
2304
960
13
39
169
1521
507
4
20
16
400
80
10
35
100
1225
350
20
63
400
3969
1260
113
419
1449
17067
4795
N = 12
a) The best fit line associated with the n points (x1, y1), (x2, y2), . . . , (xn, yn) has the form
y = b0 + b1x
where
slope = b1
=
n(xy)  (x)(y)
n(x2)  (x)2
intercept = b0
=
y  m(x)
n
So using our calculated values, we put them in formulae:
b1 =
12(4795)  (113)(419)
12(1449)  (113)2
b1 =
10193
4619
=
2.2068
And

b0 =
419-2.2068(113)
12
=
14.136
So the least-squares regression line is
y = 14.136 + 2.2068x
b) Coefficient of determination is calculated as
r
2 =
b1y + b0xy - (y)2/n
y2  (y)2/n
So we calculate it using the given data,
r2
=
14.136(419)+2.2068(4795)-(419)/12
17067-(419)2/12

r2
=
1874.51
2436.917
r2 = 0.7692
So, r2 = 0.7692 indicates that 76.92% of the variability in y is explained by its linear relationship with the independent variable x and 23.08% of the variation is due to chance or other factors.
c) The estimated number of new policies by an agent having 15 years of experience is
y = 2.2068*15+14.136
=47.2804
d) If r = 0, i.e., the coefficient of correlation is zero, than there is no linear relationship between the dependent and the independent variable. This means that the slope of least-squares regression line will change from point to point and will not be constant.
e) The coefficient of correlation is r = 0.887, this means that there is a strong positive relationship between the independent and dependant variable. Hence the least-squares regression line fitted is very good. So if we estimate the number of policies written by an agent with one year experience, it will be reliable.
f) The slope of the line, i.e., b1=2.2068, indicates that the values of y increase by 2.2068 units for a unit increase in x.
Bibliography
Chatterjee, Samprit and Hadi, Ali S. Regression Analysis by Example. 4th Edition. Wiley Series in Probability and Statistics. 2006. Read More
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