MATH1312 Regression Analysis Assignment Help

MATH1312 Regression Analysis Assignment Help

MATH1312 Regression Analysis Assignment Help

1 A.

Scatter plot:

MATH1312 Regression Analysis Assignment Help

From the above plot, it can be identified that:

The pH value of around 6.5 is the most prevalent in the different lakes irrespective of their sizes.

For smaller lakes, the pH value is lying to acidic side (i.e., having a pH value less than 7), while the lakes that are bigger are both acidic as well as basic.

The largest lake is the one at the highest pH value having an exceptionally high value for a sigficantly larger area as compared to the other lakes.

1 B.

For using the principal of least squares for fitting a regression line to the data where pH is the dependent variable (y) and area is the independent variable (x), following calculation is needed to be done:

Observation

Area (x)

ph (y)

xy

x2

y2

1

33

6.6

217.8

1089

43.56

2

161

6.4

1030.4

25921

40.96

3

189

6.5

1228.5

35721

42.25

4

149

6.9

1028.1

22201

47.61

5

47

7.1

333.7

2209

50.41

6

170

7.5

1275

28900

56.25

7

352

8.8

3097.6

123904

77.44

8

187

6.4

1196.8

34969

40.96

9

76

5.9

448.4

5776

34.81

10

52

6.7

348.4

2704

44.89

11

175

7.1

1242.5

30625

50.41

12

53

6.6

349.8

2809

43.56

13

200

8.0

1600

40000

64

Sum

1844

90.5

13397

356828

637.11

From the above table, Σx = 1844, Σy = 90.5, Σxy = 13397, Σx2 = 356828, Σy2 = 637.11 n is the sample size (13, in our case).

We will use the following formula for finding a and b, where the regression line is y = a + bx

MATH1312 Regression Analysis Assignment Help

a= 6.13, b= 0.006

Hence the regression line is y = 6.127 + 0.005x

MATH1312 Regression Analysis Assignment Help

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1C.

For the ANOVA test, following table is needed to be calculated:

 

DF

SS

MS

F

Regression

1

SSR= ∑ (yiy)2

MSR=SSR/1

F=MSR/MSE

Residual

n-2

SSE=∑(yiyi)2

MSE=SSE/(n−2)

 

Total

n-1

SSTO=∑(yiy)2

 

 

Whereyiis the estimated valued through the regression line

Following is the calculation for the table

Observation

Area (x)

ph (y)

xy

x^2

y^2

estimated y

(y-estimated y)^2

(estimated y- avg. of y)^2

(y-avg. of y)^2

1

33

6.6

217.8

1089

43.56

               6.32

                            0.08

                                            0.41

                    0.13

2

161

6.4

1030.4

25921

40.96

               7.07

                            0.46

                                            0.01

                    0.32

3

189

6.5

1228.5

35721

42.25

               7.24

                            0.55

                                            0.08

                    0.21

4

149

6.9

1028.1

22201

47.61

               7.00

                            0.01

                                            0.00

                    0.00

5

47

7.1

333.7

2209

50.41

               6.40

                            0.48

                                            0.31

                    0.02

6

170

7.5

1275

28900

56.25

               7.13

                            0.14

                                            0.03

                    0.29

7

352

8.8

3097.6

123904

77.44

               8.20

                            0.36

                                            1.53

                    3.38

8

187

6.4

1196.8

34969

40.96

               7.23

                            0.68

                                            0.07

                    0.32

9

76

5.9

448.4

5776

34.81

               6.57

                            0.46

                                            0.15

                    1.13

10

52

6.7

348.4

2704

44.89

               6.43

                            0.07

                                            0.28

                    0.07

11

175

7.1

1242.5

30625

50.41

               7.16

                            0.00

                                            0.04

                    0.02

12

53

6.6

349.8

2809

43.56

               6.44

                            0.03

                                            0.27

                    0.13

13

200

8.0

1600

40000

64

               7.30

                            0.48

                                            0.12

                    1.08

Sum

1844

90.5

13397

356828

637.11

             90.51

                            3.80

                                            3.29

                    7.09

Following is the ANOVA table:

 

DF

SS

MS

F

Regression

1

3.291011

3.291011

9.527217

Residual

11

3.799758

0.345433

 

Total

12

7.090769

 

 

We are testing the null hypothesis H0: b = 0 against the alternative hypothesis Ha: b ≠ 0.

If b = 0 then F = 1

If F > 1 then b ≠ 0, which means there is a linear relationship

In view of the fact there F = 9.52, we can say there is a linear relationship

1D.

MATH1312 Regression Analysis Assignment Help

Following are the assumptions of Regression

linearity and additivity – This can be seen from normal probability plot

statistical independence – This can be seen from the Versus fits

homoscedasticity - This can be seen from the Versus Orders

normality - This can be seen from the histogram

1E.

The regression line is y = 6.128 + 0.00588x

If x = 2050 then pH = 18.18

For developing a 99% CI for this prediction, following is the computation needed:

MATH1312 Regression Analysis Assignment Help

Computing the values, following is the confidence interval (6.88, 29.47)

2 A.

For using the principal of least squares for fitting a regression line to the data where pressure is the dependent variable (y) and steam is the independent variable (x), following calculation is needed to be done:

Observation

steam (x)

pressure (y)

xy

x^2

y^2

1

35.3

10.98

387.59

1,246.09

120.56

2

29.7

11.13

330.56

882.09

123.88

3

30.8

12.51

385.31

948.64

156.50

4

58.8

8.4

493.92

3,457.44

70.56

5

61.4

9.27

569.18

3,769.96

85.93

6

71.3

8.73

622.45

5,083.69

76.21

7

74.4

6.36

473.18

5,535.36

40.45

8

76.7

8.5

651.95

5,882.89

72.25

9

70.7

7.82

552.87

4,998.49

61.15

10

57.5

9.14

525.55

3,306.25

83.54

11

46.4

8.24

382.34

2,152.96

67.90

12

28.9

12.19

352.29

835.21

148.60

13

28.1

11.88

333.83

789.61

141.13

14

39.1

9.57

374.19

1,528.81

91.58

15

46.8

10.94

511.99

2,190.24

119.68

16

48.5

9.58

464.63

2,352.25

91.78

17

59.3

10.09

598.34

3,516.49

101.81

18

70

8.11

567.70

4,900.00

65.77

19

70

6.83

478.10

4,900.00

46.65

20

74.5

8.88

661.56

5,550.25

78.85

21

72.1

7.68

553.73

5,198.41

58.98

22

58.1

8.47

492.11

3,375.61

71.74

23

44.6

8.86

395.16

1,989.16

78.50

24

33.4

10.36

346.02

1,115.56

107.33

25

28.6

11.08

316.89

817.96

122.77

Sum

1315

235.6

11821.43

76323.42

2284.11

From the above table, Σx = 1315, Σy = 235.6, Σxy = 11821.43, Σx2 = 76323.42, Σy2 = 2284.11 n is the sample size (25, in our case).

We will use the following formula for finding a and b, where the regression line is y = a + bx

MATH1312 Regression Analysis Assignment Help

a= 13.62, b= -0.079

Hence the regression line is y = 13.62 - 0.079x

The least square estimates for constant is 13.62 and slope is  -0.079

2B.

Following are the residuals

Observation

steam (x)

pressure (y)

xy

x^2

y^2

estimated y

Residual

1

35.3

10.98

387.59

1,246.09

120.56

10.81

0.17

2

29.7

11.13

330.56

882.09

123.88

11.25

-0.12

3

30.8

12.51

385.31

948.64

156.50

11.16

1.35

4

58.8

8.4

493.92

3,457.44

70.56

8.93

-0.53

5

61.4

9.27

569.18

3,769.96

85.93

8.72

0.55

6

71.3

8.73

622.45

5,083.69

76.21

7.93

0.80

7

74.4

6.36

473.18

5,535.36

40.45

7.68

-1.32

8

76.7

8.5

651.95

5,882.89

72.25

7.50

1.00

9

70.7

7.82

552.87

4,998.49

61.15

7.98

-0.16

10

57.5

9.14

525.55

3,306.25

83.54

9.03

0.11

11

46.4

8.24

382.34

2,152.96

67.90

9.92

-1.68

12

28.9

12.19

352.29

835.21

148.60

11.32

0.87

13

28.1

11.88

333.83

789.61

141.13

11.38

0.50

14

39.1

9.57

374.19

1,528.81

91.58

10.50

-0.93

15

46.8

10.94

511.99

2,190.24

119.68

9.89

1.05

16

48.5

9.58

464.63

2,352.25

91.78

9.75

-0.17

17

59.3

10.09

598.34

3,516.49

101.81

8.89

1.20

18

70

8.11

567.70

4,900.00

65.77

8.03

0.08

19

70

6.83

478.10

4,900.00

46.65

8.03

-1.20

20

74.5

8.88

661.56

5,550.25

78.85

7.68

1.20

21

72.1

7.68

553.73

5,198.41

58.98

7.87

-0.19

22

58.1

8.47

492.11

3,375.61

71.74

8.98

-0.51

23

44.6

8.86

395.16

1,989.16

78.50

10.06

-1.20

24

33.4

10.36

346.02

1,115.56

107.33

10.96

-0.60

25

28.6

11.08

316.89

817.96

122.77

11.34

-0.26

Sum

1315

235.6

11821.432

76323.42

2284.1102

235.6

-1.6E-14

2C.

For the ANOVA table following are needed to be calculated:

Observation

steam (x)

pressure (y)

xy

x^2

y^2

estimated y

(y-estiamted y)^2

(estimated y- avg. of y)^2

(y-avg. of y)^2

Residual

1

35.3

10.98

387.59

1,246.09

120.56

10.81

0.03

1.91

2.42

0.17

2

29.7

11.13

330.56

882.09

123.88

11.25

0.02

3.35

2.91

-0.12

3

30.8

12.51

385.31

948.64

156.50

11.17

1.81

3.03

9.52

1.34

4

58.8

8.4

493.92

3,457.44

70.56

8.93

0.28

0.24

1.05

-0.53

5

61.4

9.27

569.18

3,769.96

85.93

8.72

0.30

0.49

0.02

0.55

6

71.3

8.73

622.45

5,083.69

76.21

7.93

0.63

2.22

0.48

0.80

7

74.4

6.36

473.18

5,535.36

40.45

7.69

1.76

3.02

9.39

-1.33

8

76.7

8.5

651.95

5,882.89

72.25

7.50

1.00

3.69

0.85

1.00

9

70.7

7.82

552.87

4,998.49

61.15

7.98

0.03

2.08

2.57

-0.16

10

57.5

9.14

525.55

3,306.25

83.54

9.03

0.01

0.15

0.08

0.11

11

46.4

8.24

382.34

2,152.96

67.90

9.92

2.82

0.25

1.40

-1.68

12

28.9

12.19

352.29

835.21

148.60

11.32

0.76

3.58

7.65

0.87

13

28.1

11.88

333.83

789.61

141.13

11.38

0.25

3.83

6.03

0.50

14

39.1

9.57

374.19

1,528.81

91.58

10.50

0.87

1.16

0.02

-0.93

15

46.8

10.94

511.99

2,190.24

119.68

9.89

1.11

0.22

2.30

1.05

16

48.5

9.58

464.63

2,352.25

91.78

9.75

0.03

0.11

0.02

-0.17

17

59.3

10.09

598.34

3,516.49

101.81

8.89

1.44

0.28

0.44

1.20

18

70

8.11

567.70

4,900.00

65.77

8.04

0.01

1.92

1.73

0.07

19

70

6.83

478.10

4,900.00

46.65

8.04

1.46

1.92

6.73

-1.21

20

74.5

8.88

661.56

5,550.25

78.85

7.68

1.45

3.05

0.30

1.20

21

72.1

7.68

553.73

5,198.41

58.98

7.87

0.04

2.42

3.04

-0.19

22

58.1

8.47

492.11

3,375.61

71.74

8.99

0.27

0.19

0.91

-0.52

23

44.6

8.86

395.16

1,989.16

78.50

10.06

1.45

0.41

0.32

-1.20

24

33.4

10.36

346.02

1,115.56

107.33

10.96

0.36

2.35

0.88

-0.60

25

28.6

11.08

316.89

817.96

122.77

11.34

0.07

3.67

2.74

-0.26

Sum

1315

235.6

11821.432

76323.42

2284.1102

235.638

18.2234617

45.5596905

63.8158

-0.038

 

DF

SS

MS

F

Regression

1

SSR= ∑ (yiy)2

MSR=SSR/1

F=MSR/MSE

Residual

n-2

SSE=∑(yiyi)2

MSE=SSE/(n−2)

 

Total

n-1

SSTO=∑(yiy)2

 

 

Whereyiis the estimated valued through the regression line

 

df

SS

MS

F

Regression

1

45.5924

45.5924

57.54279

Residual

23

18.2234

0.792322

 

Total

24

63.8158

 

 

The ANOVA table us used for test hypotheses about the regression linearity or population means. When the null hypothesis of equal means is true, the two mean squares estimate the same quantity (error variance), and should be of approximately equal magnitude. In other words, their ratio should be close to 1.

2D.

Following is the formula for computing correlation coefficient

MATH1312 Regression Analysis Assignment Help

Replacing the value, r= -0.845

Coefficient of determination is r2  = 0.714

2E.

STD for

Predictor

Error

Slope

Constant

Constant

-62.95

10.175

-6.142

Waist

0.874

0.108

8.123

STD

4.549

8.86

7.86

2F.

For slope

We are testing the null hypothesis H0: b = 0 against the alternative hypothesis Ha: b ≠ 0.

If b = 0 then F = 1

If F > 1 then b ≠ 0, which means there is a linear relationship

In view of the fact there F = 57.54, we can say the slope is significant

2G.

The confidence interval for Slope is (b1 is slope of the regression line, σb1is standard error estimate  (SE))

MATH1312 Regression Analysis Assignment Help

Where

SE= sqrt [ Σ(yi - ?i)2 / (n - 2) ] / sqrt [ Σ(xi - x)2 ]

Computing the value the CI is (-0.109, -0.050)

Get more Information - A Study of a 'Case Study'

3A.

Fo scatterplot of data following steps are followed in MInitab:

Choose Graph > Scatterplot.

Choose Simple, then click OK.

Under Y variables, select pressure as Y variable.

Under X variables, select steam as X variable.

Click OK.

MATH1312 Regression Analysis Assignment Help

There seems to be a negative correlation between the variables as can be seen from the scatterplot

3 B.

For linear regression, following are the steps which are taken

Choose Stat > Regression > Fitted Line Plot.

In Response (Y), enter steam.

In Predictor (X), enter pressure.

Under Type of Regression Model, choose linear.

Click OK in each dialog box.

MATH1312 Regression Analysis Assignment Help

MATH1312 Regression Analysis Assignment Help

For calculating the residuals, following are the steps:

Select Stat >> Regression >> Fit Regression Model ...

Specify the response and the predictor(s).

Under Graphs...

Under Residuals for Plots, select Regular

Under Residuals Plots, select Residuals versus fits.

Select OK.

MATH1312 Regression Analysis Assignment Help

For ANOVA table, Stat > ANOVA > General Linear Model > Fit General Linear Model

MATH1312 Regression Analysis Assignment Help

For correlation coefficient, Click “Stat”, then click “Basic Statistics” and then click “Correlation.”

MATH1312 Regression Analysis Assignment Help

Coefficient of determination is r2  = 0.714

From the output of regression:

MATH1312 Regression Analysis Assignment Help

As we can see the P value is 0 for both of the constant and slope. This means that null hypothesis for both can be rejected which stated the constant = 0 and slope – 0

From the output of regression the CI can be identified

MATH1312 Regression Analysis Assignment Help

3C.

MATH1312 Regression Analysis Assignment Help

linearity and additivity – This can be seen from normal probability plot

statistical independence – This can be seen from the Versus fits

homoscedasticity - This can be seen from the Versus Orders

normality - This can be seen from the histogram