Quantitative Methods

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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