PFA

PFA

PFA

Question No. 1

Take 30 samples (each of size 5) from the population (Do Not Submit A Printout Of These Samples)

Obtain the distribution of both S and S'(submit a printout of these two distributions)

In mathematics, the standard deviation is a positive real number, possibly infinite, used in the field of probability to characterize the distribution of a random variable around its mean. In statistics, the standard deviation or standard deviation is defined in contrast to a finite set. Tables presented below shows the standard deviation of each sample and standard deviation of the finite population.

Sample 1

Sample 2

Sample 3

Sample 4

Sample 5

Mean

4.2

3.4

5.4

6.2

4.4

N-1

4

4

4

4

4

Variance = V

23.70

7.08

15.78

21.88

4.43

Variance = V`

18.96

5.66

12.62

17.50

3.54

Standard Deviation = S

4.87

2.66

3.97

4.68

2.10

Standard Deviation = S`

4.35

2.38

3.55

4.18

1.88

Sample 6

Sample 7

Sample 8

Sample 9

Sample 10

Mean

4.2

5.8

6.8

7

4.8

N-1

4

4

4

4

4

Variance = V

4.58

15.18

18.03

16.48

15.63

Variance = V`

3.66

12.14

14.42

13.18

12.50

Standard Deviation = S

2.14

3.90

4.25

4.06

3.95

Standard Deviation = S`

1.91

3.48

3.80

3.63

3.54

Sample 11

Sample 12

Sample 13

Sample 14

Sample 15

Mean

4.6

6.2

2.8

6.6

5.6

N-1

4

4

4

4

4

Variance = V

11.68

22.68

9.63

13.18

17.63

Variance = V`

9.34

18.14

7.70

10.54

14.10

Standard Deviation = S

3.42

4.76

3.10

3.63

4.20

Standard Deviation = S`

3.06

4.26

2.78

3.25

3.76

Sample 16

Sample 17

Sample 18

Sample 19

Sample 20

Mean

4.2

6.4

7.8

5.6

5.2

N-1

4

4

4

4

4

Variance = V

11.58

18.43

26.68

19.63

18.63

Variance = V`

9.26

14.74

21.34

15.70

14.90

Standard Deviation = S

3.40

4.29

5.17

4.43

4.32

Standard Deviation = S`

3.04

3.84

4.62

3.96

3.86

Sample 21

Sample 22

Sample 23

Sample 24

Sample 25

Mean

3.4

5.6

5.8

4.8

7.4

N-1

4

4

4

4

4

Variance = V

12.78

8.83

14.18

14.03

33.08

Variance = V`

10.22

7.06

11.34

11.22

26.46

Standard Deviation = S

3.57

2.97

3.77

3.75

5.75

Standard Deviation = S`

3.20

2.66

3.37

3.35

5.14

Sample 26

Sample 27

Sample 28

Sample 29

Sample 30

Mean

4.8

3.2

6.4

4.4

5

N-1

4

4

4

4

4

Variance = V

19.63

5.93

9.23

10.63

4.58

Variance = V`

15.70

4.74

7.38

8.50

3.66

Standard Deviation = S

4.43

2.44

3.04

3.26

2.14

Standard Deviation = S`

3.96

2.18

2.72

2.92

1.91

Question 1 C

Calculate the standard error of each of the two estimators S and S' and use these values to explain whether or not your analysis supports the theory.

Sample 1

Sample 2

Sample 3

Sample 4

Sample 5

N (S) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S

2.18

1.19

1.78

2.09

0.94

N (S`) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S`

1.95

1.06

1.59

1.87

0.84

Sample 6

Sample 7

Sample 8

Sample 9

Sample 10

N (S) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S

0.96

1.74

1.90

1.82

1.77

N (S`) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S`

0.86

1.56

1.70

1.62

1.58

Sample 11

Sample 12

Sample 13

Sample 14

Sample 15

N (S) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S

1.53

2.13

1.39

1.62

1.88

N (S`) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S`

1.37

1.90

1.24

1.45

1.68

Sample 16

Sample 17

Sample 18

Sample 19

Sample 20

N (S) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S

1.52

1.92

2.31

1.98

1.93

N (S`) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S`

1.36

1.72

2.07

1.77

1.73

Sample 21

Sample 22

Sample 23

Sample 24

Sample 25

N (S) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S

1.60

1.33

1.68

1.68

2.57

N (S`) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S`

1.43

1.19

1.51

1.50

2.30

Sample 26

Sample 27

Sample 28

Sample 29

Sample 30

N (S) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S

1.98

1.09

1.36

1.46

0.96

N (S`) Sq Root

2.24

2.24

2.24

2.24

2.24

Standard Error of S`

1.77

0.97

1.22

1.30

0.86

The data analysis shows that sample has a smaller bias but a larger standard error than population data. As shown in the diagram below, standard error of the sample data is larger than the standard error than population, which is due to higher variation in the data selected as sample for the analysis. However, increase in the sample size result in dispersion of variation across larger population segment that result in reducing the standard error of the results. Therefore, theory is supported by the analysis of the data.

Question No. 2

Consider the Classical Linear Regression Model (CLRM) Y = a +ßX +( where X denotes the independent variable GDP, Y is the dependent variable HDI, a and ß are unknown constants and ( is a random variable. Use a calculator and your sample to calculate ?X, ?Y, ?XY and ?X2. Use these values to write down the pair of 'normal equations' the solutions of which give the constant term (a) and the slope coefficient (b) of the fitted Ordinary Least Squares line Y = a + bX.

S. No.

Country Name

GDP per capita in PPP terms (X)

Human Development Index (HDI) value ...