Increase of Salary of Teachers

4.1

4.

(a)

Paying teachers more does not cause the cost of prescription drugs, there might be a positive correlation and at the same time a strong correlation but this is caused by the lurking variables, this are unknown variables that will lead to strong relationship. Therefore the strong relationship between teachers pay and prescription drug cost does not mean teachers pay lead to an increase in the cost of prescription drugs, however this is a result of other variables other than teachers pay and this variables are referred to as lurking variables.

(b)

Lurking variables that may be causing the high correlation between teachers pay and prescription drugs will include inflation that is caused due to increased pay to the teachers, other variables may include higher demand for prescription drugs that lead to increase in their prices.

8.

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Increase of Salary of Teachers

(b)

The data portrays a strong negative correlation. This means that there is a strong negative correlation between variables. Negative correlation means that when one variable increases then we expect the other variable to be declining

(c)

X

Y

X

Y

XY

2

2

3

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Increase of Salary of Teachers

40

9

1600

120

7

35

49

1225

245

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Increase of Salary of Teachers

15

30

225

900

450

35

25

1225

625

875

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Increase of Salary of Teachers

75

18

5625

324

1350

TOTAL

135

148

7133

4674

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Increase of Salary of Teachers

3040

R  =                          n ∑xy – (∑x)(∑y)

____________________________

(n ∑ x2 – (∑ x)2 )½ (n∑ y2 – (∑y)2)½

 R  = 5 (3040) – (135)( 148)

____________________________

(5 (7133)– (135)2 )½ (5(4674) – (148)2)½

R = -0.94534

From the results of the correlation coefficient it is clear that the variables have a strong negative relationship, the negative relationship means that as the value of x increases then the value of Y will decline, this is due to their negative relationship given by the correlation coefficient.

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12.

(b)

The correlation is moderate. However the correlation is positive as depicted by the scatter diagram.

(c)

X

Y

X

Y

XY

2

2

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12.5

26

156.25

676

325

30

73

900

5329

2190

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Increase of Salary of Teachers

24.5

39

600.25

1521

955.5

14.3

23

204.49

529

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Increase of Salary of Teachers

328.9

7.5

15

56.25

225

112.5

27.7

30

767.29

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Increase of Salary of Teachers

900

831

16.2

15

262.44

225

243

20.1

25

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Increase of Salary of Teachers

404.01

625

502.5

total

152.8

246

3350.98

10030

5488.4

R  =                          n ∑xy – (∑x)(∑y)

____________________________

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Increase of Salary of Teachers

(n ∑ x2 – (∑ x)2 )½ (n∑ y2 – (∑y)2)½

 R  = 8 (5488.4) – (152.8)( 246)

____________________________

(8 (3350.98)– (152.8)2 )½ (8(10030) – (246)2)½

R = 0.764842

From the results of our calculated r it is clear that there is a moderate positive correlation between the two variables, therefore an increase in x will result into an increase in Y.

16.

(a) r = 0.820 significant at the a = 0.01 level sample size n = 7 and n = 9

n

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Increase of Salary of Teachers

α

critical  value

r

significance

7

0.01

0.87

0.82

insignificant

9

0.01

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Increase of Salary of Teachers

0.8

0.82

significant

When we consider the sample size which is 7 then we conclude that the correlation coefficient is not significant because the value from the table is greater than our calculated correlation, when we consider a larger sample size where n = 9 then we conclude that the correlation coefficient is significant where the value from the table is less than the value calculated for correlation.

(b) r = 0.40, a = 0.05 level sample size n = 20 data pairs and n = 27

n

α

critical  value

r

significance

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Increase of Salary of Teachers

20

0.05

0.44

0.4

insignificant

27

0.05

0.38

0.4

significant

When we consider the sample size 20 then we conclude that the correlation coefficient is not

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Increase of Salary of Teachers

significant because the value from the table is greater than our calculated correlation, when we consider a larger sample size where n = 27 then we conclude that the correlation coefficient is significant where the value from the table is less than the value calculated for correlation.

(c)

For r value to be significant then we have to consider the value of r and also the level of test and the value they provide at the test level, when our r value is greater than the critical value given then r is significant but if it is less then our r is not significant. However from our above examples the greater the sample size then the more significant our r will be, therefore when we use a larger sample size the higher the possibility that the r value will be significant.

4.2

4.

(b)

x

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Increase of Salary of Teachers

y

x2

y2

xy

0

50

0

2500

0

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Increase of Salary of Teachers

2

43

4

1849

86

5

33

25

1089

165

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Increase of Salary of Teachers

6

26

36

676

156

total

13

152

65

6114

407

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Increase of Salary of Teachers

R  =                          n ∑xy – (∑x)(∑y)

____________________________

(n ∑ x2 – (∑ x)2 )½ (n∑ y2 – (∑y)2)½

 R  = 4 (407) – (13)( 152)

____________________________

(4 (65)– (13)2 )½ (4(6114) – (152)2)½

R= -0.99213

(c)

Y = a + b X

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B =                         n ∑xy – (∑x)(∑y)

______________________

n ∑ x2 – (∑ x)2

 B = 4 (407) – (13)(152)

______________________

4 (65) – (13)2

B = -3.82

A = y’ – x’b

A = 50.42

Y = 50.42 – 3.82 X

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(d)

Regression line:

(e)

R2 correlation of determination:

R2 = 0.98

This means that 98% of variations in Y can be explained by X, however 1.2% variations in Y is not explained by X.

(f)

If x = 4 then Y = 35.13187

6.

(b)

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Increase of Salary of Teachers

x

y

x2

y2

xy

37

5

1369

25

24/40

Increase of Salary of Teachers

185

47

8

2209

64

376

57

10

3249

25/40

Increase of Salary of Teachers

100

570

67

16

4489

256

1072

77

30

26/40

Increase of Salary of Teachers

5929

900

2310

87

43

7569

1849

3741

total

372

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Increase of Salary of Teachers

112

24814

3194

8254

mean

62

18.66667

 R  = 6 (8254) – (372)( 112)

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Increase of Salary of Teachers

____________________________

(6 (24814) – (372)2 )½ (6(3194) – (112)2)½

R = 0.942755

(c)

Y = a + b X

 B = 6 (8254) – (372)( 112)

______________________

6 (24814) – (372)2

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Increase of Salary of Teachers

B = 0.748

A = y’ – x’b

A = -27.74

Y = -27.74 + 0.748 X

(e)

R2 correlation of determination:

R2 = 0.8887

This means that 88.87% of variations in Y can be explained by X, however 11.13% variations in Y is not explained by X.

(f)

If x = 70 then our y will be 24.65524, this is in reference to our estimated regression model.

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Increase of Salary of Teachers

Chapter review:

(b)

x

y

x2

y2

xy

4

33

16

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Increase of Salary of Teachers

1089

132

7

37

49

1369

259

5

34

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Increase of Salary of Teachers

25

1156

170

6

32

36

1024

192

1

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Increase of Salary of Teachers

32

1

1024

32

5

38

25

1444

190

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Increase of Salary of Teachers

9

43

81

1849

387

10

37

100

1369

370

35/40

Increase of Salary of Teachers

10

40

100

1600

400

3

33

9

1089

36/40

Increase of Salary of Teachers

99

total

60

359

442

13013

2231

mean

6

35.9

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Increase of Salary of Teachers

Y = a + b X

 B = 10 (2231) – (60)(359)

______________________

10 (442) – (60)2

B = 0.939

A = y’ – x’b

A = 30.26585366

Y = 30.26 + 0.939X

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Increase of Salary of Teachers

Regression line:

(c)

Correlation coefficient r = 0.760856

Correlation of determination = 0.5789

Therefore 57.89% of variations in Y are explained by x, while 42.11% of variations in Y are not explained by X.

(d)

X = 2

Using the regression estimates then the value of annual salary(Y) predicted by the regression is 32.1439

## References:

A. Bluman (2001) Elementary Statistics: AStep by Step Approach, McGraw Hill publishers, New York

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