Maths

Maths

Question 1

1. (i) Use the data in this file “Censusdata.xls” to develop quotas for this survey by age and gender for one of the regions**. Briefly explain how you calculated the quotas.

We choose Dublin city as our region for analysis, first we get the quotas for the region by age as follows:

First we add up the total for the regions, this is to say we add the males and females at their age groups as follows:

total male and female

percentage

0-9 years

51406

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10%

10-19 years

54297

11%

20-29 years

119454

24%

30-39 years

84043

17%

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40-49 years

62527

12%

50-59 years

49866

10%

60-69 years

38851

8%

70-79 years

30064

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6%

80 years+

15703

3%

total

506211

After getting the total we divide each total of males and females with the sum total of the region and then get the percentage level of this ratio.

Quotas by gender:

For the selected sample of Dublin city it is easy to get the quota of males and females, this is done by simply adding up all the females and the males as follows then getting the percentage value of each as follows:

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total

percentage

male

248087

49%

female

258124

51%

total

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506211

1. (ii) What other variables do you think the quotas should be broken down by for this survey

In this region the survey can be broken down into regions, this can consider the quota by the regions that are given.

1. (iii) Explain briefly why you feel quota sampling may or may not be suitable for this survey

Quota sampling is one of the methods that can be used to obtain a sample from a population, the greatest weakness of this type of sampling is that it is not randomly selected and therefore the issue of probability in choosing an element for the sample is not used. For this survey therefore it would not be appropriate to use quota sampling as it does not use random selection of elements of the sample.

Question 2

(i)

The best technique to select a sample is through a random sample, random sampling involves the use of random numbers to select the sample. The first step is to assign a numbers to the sample, this involves for example assigning each company from number 1 to the population size number, in our case we assign the numbers 1 to 50.

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The second step is to come up with random numbers, this random numbers can be generated using a calculator or computer, the random numbers produced will in our case be a two number random number and for this reason we will consider the first two numbers of every random number produced, for example when we get the random number 0.3899003 then we will consider 0.38 which will give us the random number 38, if we get a random that exceeds our sample size for example when we have 0.9877 then we reject this random number.

When 20 valid random samples are produced the next step is to match the random samples to the sample, this involves the recording of the sample by selecting those elements of the data that correspond to the random number generated. The results from these steps give us the random sample.

In our case we will have the following random numbers generated using a computer,

31, 6, 30, 27, 13, 4, 5, 38, 46, 47, 42, 26, 32, 37, 25, 36, 34, 17, 3 and 10

The above random numbers correspond to the following elements in the data as shown in the table below which shows the random sample collected:

sample

random  numbers

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value

1

31

company

31

36000

2

6

company

6

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23000

3

30

company

30

29000

4

27

company

27

3000

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5

13

company

13

49000

6

4

company

4

41000

7

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5

company

5

46000

8

38

company

38

20000

9

46

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company

46

40000

10

47

company

47

30000

11

42

company

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42

13000

12

26

company

26

7000

13

32

company

32

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1000

14

37

company

37

45000

15

25

company

25

22000

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16

36

company

36

17000

17

34

company

34

6000

18

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17

company

17

34000

19

3

company

3

38000

20

10

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company

10

24000

(ii) Using the sample of 20 tax returns calculate

sales

1

36000

2

23000

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3

29000

4

3000

5

49000

6

41000

7

46000

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8

20000

9

40000

10

30000

11

13000

12

7000

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13

1000

14

45000

15

22000

16

17000

17

6000

18

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34000

19

38000

20

24000

total

524000

mean

26200

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The total for the selected sample is 524000, to get the mean of this data we divide the total by the sample size which in this case will be 524,000 / 20 = 26200 this means that the mean or average is equal to 26200.

The standard deviation is derived from the following formula: the formula below gives the variance and to get the standard deviation we find the square root of the variance:

∑ (X –X’)2

S2 = ____________

N -1

Where X’ is the mean and x is the sample elements, N is the sample size which in this case is equal to 20, the table below shows the calculation of the standard deviation

sales

X -X’

(X-X’)2

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1

36000

9800

96040000

2

23000

-3200

10240000

3

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29000

2800

7840000

4

3000

-23200

538240000

5

49000

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22800

519840000

6

41000

14800

219040000

7

46000

19800

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392040000

8

20000

-6200

38440000

9

40000

13800

190440000

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10

30000

3800

14440000

11

13000

-13200

174240000

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12

7000

-19200

368640000

13

1000

-25200

635040000

14

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45000

18800

353440000

15

22000

-4200

17640000

16

17000

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-9200

84640000

17

6000

-20200

408040000

18

34000

7800

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60840000

19

38000

11800

139240000

20

24000

-2200

4840000

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total

524000

4.273E+09

mean

26200

2.249E+08

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standard deviation

14996.84

14996.842

4.273 X 109

S2 = ____________

20 -1

S2 = 2.249 X 108

S = 14996.842

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Quartile range:

The range of the data is given by the maximum amount minus the minimum amount, in our case the minimum amount is 1000 and the maximum amount is 49000, therefore our range is equal to 49,000 -1,000 = 48,000

From our sample that was generated randomly our mean sales value is 26200 with a standard deviation value equal to 14996.84 and our range of this data is equal to 48,000. this shows that we have a very large deviation of data from the mean as depicted by the standard deviation and the range.

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