Quantitative methods 2
Question 1.
Two samples of data selected at random from the payroll file held on the computer of a medium sized company provided the following typical weekly salary, in £s, for the Admin
question one
group A
group S
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119
117
148
94
154
97
124
138
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118
136
130
124
140
131
152
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134
140
123
161
145
185
90
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134
126
152
99
145
89
158
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92
111
105
173
154
173
156
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167
152
172
115
146
107
154
138
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141
146
119
128
167
108
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101
89
162
96
101
107
165
103
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151
148
total
4363
3587
mean
145.4333
119.5667
standard deviation
22.16036
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21.78212
median
149.5
120
minimum value
101
89
1st quartile
131
100
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2nd quartile (median)
149.5
120
3rd quartile
161.75
137.5
maximum value
185
156
The total for group S is 3587 and therefore the mean is 119.5667, the standard deviation for this data is 21.78212, the median is equal to 120, minimum value of the data is 89, the 1st quartile is
100, 2 nd quartile or the median is 120, third quartile is 137.5 and finally the maximum value is 156.
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The total for group A is 4363 and therefore the mean is 145.4333, the standard deviation for this data is 22.16, the median is equal to 149.5, minimum value of the data is 101, the 1st quartile is
131, 2 nd quartile or the median is 149.5, third quartile is 161.75 and finally the maximum value is 185.
Produce Box-and-whisker plot for each data set and write a short explanation about the distribution of the data sets.
From the above group A box and wisker diagram the data given seems to be negatively skewed, this is becosue of the fact that the left hand side of the diagram extends more than the right hand side, this means that more observations are on the low end of data measures,therefore the data is negatively skewed according to the box and whisker diagram.
From the above box and whisker diagram of group S then we can conclude that the data is positively skewed, this is evident from the maximum value having extended more than the left hand side, therefore the data is positively skewed meaning that more observations are in the higher values of measure.
Question 2.
A company employs a number of skilled, semi-skilled and low skilled workers in its workshops. The rate of pay (£/hr) of each employee is determined by the level of skill required for the job and is graded from A to F. The number of employees in the three-year period from 2004 to 2006, together with the rate of pay for each employment grade is given below.
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2004
2005
2006
Rate
Number
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Rate
Number
Rate
Number
A
6.2
12
8.2
8
10.5
7
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B
8.5
6
7.6
6
9
8
C
9.6
10
9.6
9
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9.5
5
D
4.8
15
7.2
12
6.55
18
E
6.1
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20
5.9
30
7
36
F
3.35
28
5.4
30
3.55
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35
Using 2006 as base year, Calculate and compare the simple price index for grade B and D rate of pay;
2004
2005
2006
B
0.944444
0.844444
1
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D
0.732824
1.099237
1
From the table we assume that the 2006 rates are 100 percent and for this reason we divide the rates for the other years by the 2006 rates, the table above summarises the rates for group B and D for the years 2004 and 2005.
Calculate the Laspeyres and Paasche price indices for all grades and give a brief comment about the results you obtained
paasche index
2005
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2005
2006
2006
Pn X Q n
Po X Qn
Paasch index
Pn X Q n
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Po X Qn
Paasche index
A
65.6
49.6
132%
73.5
43.4
169%
B
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45.6
51
89%
72
68
106%
C
86.4
86.4
100%
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47.5
48
99%
D
86.4
57.6
150%
117.9
86.4
136%
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E
177
183
97%
252
219.6
115%
F
162
100.5
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161%
124.25
117.25
106%
The above table represents the Paasche price indix for the year 2005 and 2006, this index is calculated by multiplying the price at the curetn period by the quantity at the current perios and this total is divided by the price at the base year multiplied by the quanityt at the current period.
laspeyres index
2005
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2005
2006
2006
Pn X Q0
P0XQ0
lasp index
Pn X Q0
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P0XQ0
lasp index
A
98.4
74.4
1.322580645
126
74.4
1.693548
B
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45.6
51
0.894117647
54
51
1.058824
C
96
96
1
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95
96
0.989583
D
108
72
1.5
98.25
72
1.364583
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E
118
122
0.967213115
140
122
1.147541
F
151.2
93.8
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1.611940299
99.4
93.8
1.059701
The table above represents the laspeyres index, it is derived from multiplying the price at the current period by the quantity at the base year and the total is divided by the price at the baser year multiplied by the quantity at the base year.
Question 3.
The following table gives the average number of new cars bought, in 000s, between 1970 and 1992 along with the average disposable income, in £s, of the people who bought the cars.
Year
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Number of new cars Y
income X
1970
91.4
3912
1971
108.5
3940
1972
177.6
4256
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1973
137.3
4523
1974
102.8
4487
1975
98.6
4514
1976
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106.5
4501
1977
109.4
4409
1978
131.6
4734
1979
142.1
4998
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1980
126.6
5067
1981
124.5
5025
1982
132.1
5004
1983
150.5
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5133
1984
146.6
5309
1985
153.5
5472
1986
156.9
5703
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1987
168
5882
1988
184.2
6221
1989
192.1
6506
1990
167.1
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6621
1991
133.3
6561
1992
133.3
6717
Using EXCEL produce a scatter plot of :
Y and X;
From the chart above it is clear that as the level of income increases then the number of cars bought increases, therefore the sign of correlation in this case is positive, meaning that as one variable increases then the other variable also increases, the line above is the trend line in the scatter diagram.
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Year and Y
The chart above represents the number of cars versus year, from the chart it is clear that as the years progress the level of income also increases, the trend line shows that the correlation sign in this case is positive.
Year and X
The chart above represents the income level versus the years, the trend line shows that over the years the level of income has increased gradually and therefore there is positive correlation.
(b) Perform a linear regression analysis of the data designed to estimate the number of new cars for given disposable income and interpret the regression coefficients.
In this case our estimated model will be to find out how the level of income affects the number of cars bought; therefore number of cars is the dependent variable while the income level is the dependent variable:
Year
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Number of new cars Y
income X
Y2
X2
YX
1970
91.4
3912
8353.96
15303744
357556.8
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1971
108.5
3940
11772.25
15523600
427490
1972
177.6
4256
31541.76
18113536
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755865.6
1973
137.3
4523
18851.29
20457529
621007.9
1974
102.8
4487
10567.84
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20133169
461263.6
1975
98.6
4514
9721.96
20376196
445080.4
1976
106.5
4501
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11342.25
20259001
479356.5
1977
109.4
4409
11968.36
19439281
482344.6
1978
131.6
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4734
17318.56
22410756
622994.4
1979
142.1
4998
20192.41
24980004
710215.8
1980
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126.6
5067
16027.56
25674489
641482.2
1981
124.5
5025
15500.25
25250625
625612.5
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1982
132.1
5004
17450.41
25040016
661028.4
1983
150.5
5133
22650.25
26347689
772516.5
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1984
146.6
5309
21491.56
28185481
778299.4
1985
153.5
5472
23562.25
29942784
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839952
1986
156.9
5703
24617.61
32524209
894800.7
1987
168
5882
28224
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34597924
988176
1988
184.2
6221
33929.64
38700841
1145908
1989
192.1
6506
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36902.41
42328036
1249803
1990
167.1
6621
27922.41
43837641
1106369
1991
133.3
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6561
17768.89
43046721
874581.3
1992
133.3
6717
17768.89
45118089
895376.1
total
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3174.5
119495
455446.8
6.38E+08
16837081
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N
23
∑YX
16837080.6
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∑X2
637591361
∑Y2
455446.77
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∑X
119495.0000
∑Y
3174.5
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B
7915976.3
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385546278
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0.02053184
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A
31.34987909
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Our estimated model is
Y = 31.35 + 0.02053184 X
Where Y is the number of cars bought and X is the income level, to interpret this model we can state that an increase in income by one unit will increase the number of cars bought by 0.0205 holding all other factors constant. Also if we assume that the level of income is zero and all other factors are held constant then the number of cars bought will be 31.35.
(c) From the analyses above, obtain the coefficient of correlation between the number of new cars and disposable income and interpret.
The correlation coefficient for the number of cars and income is 0.639175237, the coefficient is positive and for this reason as one variable increases then the other variable also increases, the number is also close to the number one which shows perfect correlation, therefore we can conclude that there is a strong relationship between the two variables.
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(d) Using a suitable equation obtained from the regression analysis above, estimate the number of new cars bought for the following disposable income:
(i) £2,879; (ii) £7,489; (iii) £ 5,590
income
constant
slope
BX(slope X income)
total number of cars
2879
31.34987909
0.020532
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59.11118
90.4610599
7489
31.34987909
0.020532
153.763
185.112864
5,590
31.34987909
0.020532
114.773
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146.122891
When the income level is 2,879 then the number of cars bought is 90.46, this is 90 cars, when the income level is 7489 then the number of cars bought is 185.122 which is 185 cars and finally when the income level is 5,590 then the number of cars bought is 146.122 which is 146 cars.
(e) Comment briefly on the likely accuracy of your results for (d).
From the results above it is clear that as income increases then the number of cars bought will also increase, the model estimated can be used to predict the number of cars bought given income but this will depend on the level of the stochastic variable which is the error term, therefore the accuracy of our model will depend on the error term associated with the model and also the standard errors of our variables.
F. Comment briefly on the relevance of disposable income, as the only factor in predicting the number of new cars bought.
Income is relevant in the determination of the number of cars purchased, this is becosue as the level of income increases then the consumers will have a larger disposable income which will increase their demand for goods and one of these goods wil be new cars, however there are
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other factors that influence the purchase of new cars but to this model the assumption was that only one factor influences the number of cars bought, some of the factors that would have been considered include the price of the cars.
References:
Bluman A. (2000) Elementary Statistics: A Step by Step Approach, McGraw Hill press, New York
D. Bridge (1993) Statistics: An Introduction to Quantitative Economic Research, Rand McNally publishers, Michigan
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