1. The following table shows the amount spent on advertising and the corresponding
sales of the product from 10 companies
Company
A
B
C
D
E
F
G
H
I
J
x
y
25
35
29
24
38
12
18
27
17
30
8
12
11
5
14
3
6
8
4
9
(a) Plot a scatter diagram showing the relationship between advertising cost and sales of
the product.
(b) Identify the dependent and independent variables
(c) Calculate the equation of the regression line of sales on advertising costs.
(d) Use the regression line to forecast sales if advertising costs were £1000.
(e) Determine coefficient of determination and interpret
a)
The relationship between two variables can be examined graphically using a scatter diagram. A
scatter diagram is a simple two-dimensional graph of the values of the dependent variable and
the independent variable.
The important thing to remember when drawing a scatter diagram is that the dependent
variable is always drawn on the vertical axis of the diagram and that the independent variable is
always drawn on the horizontal axis of the diagram. The dependent variable is usually
represented by Y and the independent variable is usually represented by X.
The following scatter diagram shows the volume of sales against advertising cost for the
advertising cost data
40
35
Sales ('000s)
30
25
20
15
10
5
0
0
5
10
15
Advertising cost ('00s)
B)
The regression model to be developed will relate the volume of sales to advertising cost. the
volume of sales to depend on the cost of advertising, so sales to be the dependent variable and
advertising cost to be the independent variable.
C)
A linear regression model is based on the linear function
Predicted Y = b0 + b1X
The parameter b0 is called the intercept and the parameter b1 is called the slope of the
regression line. The value of the intercept determines the point where the regression line
meets the Y axis of the graph. The value of the slope represents the amount of change in Y
when X increases by one unit.
Another name frequently used in regression analysis to refer to the independent variable is
predictor. Note that the above regression model uses only one predictor and is therefore called
a simple regression model.
In order to develop a linear regression model of the form Predicted Y = b 0 + b1X we need to
calculate the values of b0 and b1. These values are given by the following relations:
nXY - XY
b1 =
--------------------nX2-(X)2
Y
b0 =
X
---- - b1 ----n
n
Application of the above formulae to the advertising cost data will produce the following
results:
10 x 2289 – (80 x 255)
b1 =
----------------------------(10 x 756-(80)2)
22890 – 20400
=
----------------------(7560 – 6400)
2490
=
----------1160
=
2.1466
255
b0 =
80
------ - 2.1466 -----10
10
=
25.5 – 2.1466 x 8
=
8.3272
Substituting the above values in relation will get the following regression equation:
Predicted Y = 8.3272 + 2.1466X
The value of b0 is 8.3272, which means that the regression line cuts the vertical axis of the
graph at that point. Similarly, the value of b1 is 2.1466 indicating that the value of Y will increase
by 2.1466 every time that the value of X increases by 1 (obviously, when X=0, Y=8.3272).
d)
Substituting 1 (1000) in the equation for advertising cost.
Y = 8.3272 + 2.1466X
Predicted Y = 8.3272 + 2.1466 (1) = 10.48
e)
Advertising
cost £
(‘000)
Company
Sales £
(‘000)
X
Y
A
8
25
XiX
0
B
12
35
C
11
D
(Yi-Ŷ)
(Yi-Ŷ) 2
(Yi-Y) 2
-0.5
Ŷ = b0
+b1X
25.5
-0.5
0.25
0.25
4
9.5
34.084
0.916
0.84
90.25
29
3
3.5
31.938
-2.94
8.63
12.25
5
24
-3
-1.5
19.062
4.938
24.38
2.25
E
14
38
6
12.5
38.376
-0.38
0.14
156.25
F
3
12
-5
13.5
14.77
-2.77
7.67
182.25
G
6
18
-2
-7.5
21.208
-3.21
10.29
56.25
H
8
27
0
1.5
25.5
1.5
2.25
2.25
I
4
17
-4
-8.5
16.916
0.084
0.01
72.25
J
9
30
1
4.5
27.646
2.354
5.54
20.25
ƐX = 80
ƐY =
255
Ɛ (Yi-Ŷ)2
= 60.01
Ɛ (Yi-Y)2 =
594.5
Yi-Y
r2=SSR/SST
r2 = 534.49/ 594.5
∴Coefficient of Determination (r2) = 0.899
r = 0.948 (-1 < r < +1)
Interpretation: Since the R squared value is closer to 1 there is a high probability that the X is
having a high co relation and dependency on Y, due to which sales is highly dependent on
Advertising element.
2)
2. An investor is concerned with the market return for the coming year, where the
market return is defined as the percentage gain ( or loss, if negative) over the year. The
investor believes there are five possible scenarios for the national economy in the coming
year- rapid expansions, moderate expansions, no growth, moderate contractions and serious
contraction. Furthermore, she has used all of the information available to her to estimate
that the market returns for these scenarios are, respectively 23%, 18%,15%,9% and 3% .Also
she has assessed the probabilities of these outcomes are 0.12, 0.40,0.25, 0.15 and 0.08. Use
this information to describe the probability distribution of the market return. Calculate,
average return, standard deviation and variance of the probability distribution of the market
return for the coming year.
probabily return
0.12
0.4
0.25
0.15
0.08
average
return
p*R
23
18
15
9
3
2.76
7.2
3.75
1.35
0.24
15.3
deviation from expected value squared
p*SQUARED
-7.7
59.29
7.1148
-2.7
7.29
2.916
0.3
0.09
0.0225
6.3
39.69
5.9535
12.3
151.29
12.1032
sigma
square
28.11
sigma
5.301886
Average return:
=0.12*23+0.4*18+0.25*15+0.15*9+0.08*3
=15.3
Average return is 15.3%
Variance then weights each squared deviation by its probability, giving us the following
calculation:
0.12*59.29+0.4*7.29+0.25*0.09+0.15*39.69+0.08*151.29
=28.11
Standard deviation=square root of variance=28.111/2 =5.30