The Growth Of Salmon Fish Biology Essay

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Generally, there are two groups of salmon. One is the Atlantic salmon. It lives in the North Atlantic Ocean between North America and Europe. The others are species of Pacific salmon that live in the North Pacific Ocean. Like the Atlantic salmon, they live in the ocean and also in the rivers of western North America and eastern Asia. The salmon is an anadromous (uh-NAD-droh-muhss) fish. This means that it spawns in freshwater but spends much of its life at sea. When an Atlantic salmon reaches the age of two, it leaves its home in the North Atlantic. It begins a migration to the same place in the river or stream where it was born. It spawns and then returns to the ocean for two years. After building up its strength, it leaves the ocean and returns to the rivers to spawn yet again.

The growth of salmon fish scale has been studied and their growth were measured by width for first year when they were staying in ocean environment. The salmon fish scales have been enlarged 100 times, so that the evaluations are made in hundredths of an inch. In this assignment, we were given a task that consists of a set of measurement which was gathered by the Alaska Department of Fish and Game as given in Table A. (Courtesy of K. Jensen and B. Van Alen.) The content of assignment is to conduct an analysis on a scatter diagram, a fitted line and a determination if β1 differs from zero. Furthermore, we were asked to find a 95% of confidence interval for the population mean when the freshwater growth is 100.Three types of regression analyses have been completed in this assignment, namely all salmon fish regression analysis, male salmon fish regression analysis, and last but not least, female salmon fish regression analysis. All the analysis shows the regression of the marine growth over the freshwater growth.

Empirical Models

Today, many problems in engineering and the science involve a study or analysis of the relationship between or more variables. There are two types of empirical models that are deterministic and not deterministic. Deterministic models are those are able to predict the displacement perfectly, such as the pressure of a gas container is related to the temperature and yet most of the situations are deterministic in real word analysis. Therefore, there is a nondeterministic manner called regression model is used to model and explore the relationships between these variables. However, there are some assumptions need to be made in our analysis for the empirical models that we are going to use. We assume that there is only independent or predictor variable x and the relationship with the response y are linear. The data collection of the data are represented using a scatter diagram which is a graph on which each (xi,yi) pair is represented as a point plotted in a two-dimensional coordinate system. Base on the scatter diagram, it is probably reasonable to assume that the mean of the random variable Y is related to x by the following straight-line relationship:

where the slope and intercept of the line are called regression coefficients. Since the mean of Y is not exactly a linear function of x, thus it's more appropriate to express it as in the following equation:

, where is the random error term.

This model is also known as the simple linear regression model, because it has only one independent variable or regressor. Since there is no theoretical knowledge of the relationship between x and y, and the choice of the model is based on inspection of a scatter diagram, therefore it can be said that the regression model as an empirical model.

Simple Linear Regression

The case of simple linear regression considers a single regressor variable or predictor variable x and a dependent or response variable Y. Thus, Y can be described by the model

where is a random error with mean error and (unknown) variance . The random errors corresponding to different observations and are also assumed to be uncorrelated random variables. Suppose that we have n pairs of observations, an estimated regression line with the "best fit" is needed to be drawn in the scatter diagram. The German scientist Karl Gauss(1777 - 1855) proposed estimating the parameters and in Equation 1-1 to minimize the sum of the squares of the vertical deviation. We call this criterion for estimating the regression coefficients the method of least squares. Using Equation 1-2, we may express the n expressions in the sample as

(1-3)

and the sum of the squares of the deviations of the observations from the true regression line is

(1-4)

The least squares estimates of the intercept and slope in the simple linear regression model are

(1-5)

(1-6)

where and .

The fitted or estimated regression line is therefore

(1-7)

Note that each pair of observations satisfies the relationship

where is called the residual. The residual describes the error in the fit of the model to the ith observation. The residuals are used to obtain the information about the adequacy of the fitted model.

Notational, it is occasionally convenient to give special symbols to the numerator and denominator of Equation 1-6. Given data ,,… ,, let

(1-8)

and

(1-9)

Another unknown parameter in the regression model, (the variance of the error term, ). The residuals are used to obtain an estimate of . The sum of squares of the residuals, or known as the error sum of squares, is

(1-10)

It's then showed that the expected value of the error sum of squares is . Therefore an unbiased estimator of is

(1-11)

By substituting into Equation 1-10, and simplifying,

(1-12)

where is the total sum of squares of the response variable y.

Hypothesis Tests in Simple Linear Regression

3.3.1 Use of t-Tests

Suppose we wish to test the hypothesis that the slope equals a constant, . The appropriate hypotheses are

Ho:

H1: (1-13)

where we assumed a two-sided alternative. Since the errors are , it follows directly that the observations Yi are . Now is a linear combination of independent normal random variables, and consequently, is , using the bias and variance properties of the slope discussed earlier. Besides, has a chi-square distribution with n-2 degree of freedom, and is independent of . As a result of those properties, the statistic

(1-14)

follows the t-distribution with n -2 degree of freedom under Ho: . We would reject Ho: if

(1-15)

A similar procedure can be used to test hypotheses about the intercept. To test

Ho:

H1: (1-16)

We would use the statistic

(1-17)

and reject the null analysis if computed value of this test statistic, , is such that .

A very important special case of hypotheses of Equation 1-13 is

Ho:

H1: (1-18)

These hypotheses relate to the significance of regression. Failure to reject Ho: is equivalent to concluding that there is no linear relationship between x and Y.

3.3.2 Analysis of Variance Approach to Test Significance of Regression

To test significance of regression, the analysis of variance can be used. The procedure partitions the total variability in the response variable into meaningful components as the basis for the test. The analysis of variance identity is as follows:

(1-19)

The two components on the right-hand-side of Equation 1-19 measure, respectively, the amount of variability in accounted for by the regression line and the residual variation left unexplained by the regression line. Using the Equation 1-10 (the error sum of squares) and the the regression sum of squares, we can compute the following equation:

(1-20)

where is the total corrected sum of squares of y.

Since , and that and are independent chi-square random variables with n-2 and 1 degrees of freedom, respectively. Therefore, if the null hypothesis Ho: is true, the statistic,

(1-21)

follows the distribution, and we would reject Ho if . The quantities and are called mean squares. This test procedure is usually arranged in an analysis of variance table.

Confidence Intervals

3.4.1 Confidence Intervals on Slope and Intercept

Under the assumption that the observations are normally and independently distributed, a confidence interval on the slope in simple linear regression is

(1-22)

Similarly, a confidence interval on the intercept is

(1-23)

3.4.2 Confidence Intervals on the Mean Response

A confidence interval about the mean response at the value of , say , is given by

(1-24)

where is computed from the fitted regression model.

Adequacy of the Regression Model

Fitting a regression model requires several assumptions. Estimation of the model parameters requires the assumption that the errors are uncorrected random variables with mean zero and constant variance. Tests of hypotheses and interval estimation require that the errors be normally distributed. In addition, we assume that the order of the model is correct; that is, if we fit a simple linear regression model, we are assuming that the phenomenon actually behaves in a linear or first-order manner.

3.5.1 Residual Analysis

The residuals from a regression model are , where is actual observation and is the corresponding fitted value from the regression model. Analysis of the residuals is frequently helpful in checking the assumption that the errors are approximately normally distributed with constant variance, and in determining whether additional terms in the model would be useful.

As an approximate check of normality, the experimenter can construct a frequency histogram of the residuals or a normal probability plot of residuals. We may also standardize the residuals by computing. If the errors are normally distributed, approximated 95% of the standardized residuals should fall in the interval (-2, +2). Residuals that are far outside this interval may indicate the presence of an outlier, that is, an observation that is not typical of the rest of the data. Various rules have been propose for discarding outlier, but outliers sometimes provide important information about unusual circumstances of interest to experimenters and therefore should not be automatically discarded.

3.5.2 Coefficient of Determination (R2)

The coefficient of determination is

(1-25)

The coefficient is often used to judge the adequacy of a regression model. However the statistic R2 should be use with caution, because it is always possible to make R2 unity by simply adding enough terms to the model. For example, we can obtain a "perfect" fit to n data points with a polynomial of degree n - 1. In addition, R2 will always increase if we add a variable to the model, but this does not necessarily imply that the new model is superior to the old one. Unless the error sum of squares in the new model is reduced by an amount equal to the original error mean square, the new model will have a larger error mean square than the old one, because of the loss of one error degree of freedom. Thus, the new model will actually be worse than the old one.

4.0 METHODS

In this section, we discuss about the methods used to perform regression analysis. The methods used to compute all the results are shown. Microsoft Office Excel 2007 is used as our main analyses software to conduct the analysis. Our analyses include obtaining scatter diagram, linear fitted line, test hypotheses and the confidence interval. All these analyses can be done by using one of the functions of Microsoft Excel that is 'Regression' in 'Data Analysis'.

4.1 Steps of Using 'Data Analysis' in Microsoft Excel

First of all, there is an important step to do before we can start to analyze our data that is make sure the availability of function 'Data Analysis' in our Microsoft Excel. 'Data' from the menu tab is chosen to check the availability of the function. If 'Analysis' column is not found, then add in function is needed to obtain the function of data analysis. Function 'Data Analysis' can be retrieved by customizing the quick access toolbar using the tabs of 'Add-Ins'. After finish the 'Add- Ins' process, then only we can proceed to analyze our data.

Then, 'Data Analysis' is clicked and an instruction box (Data Analysis) is popped out. Next, 'Regression' is chosen and another instruction box (Regression). We started our input process by first input the 'Input Y Range', which the data of dependent / response variables is keyed in; and second is 'Input X Range', which the data of independent / regressor variables is keyed in. The 'Input Y Range' is the data of First Year Marine Growth, while the 'Input X Range' is the data of Freshwater Growth. For our assignment, only 'Label' and 'Line Fit Plot' is ticked and any of the output options can be selected based on the users. Lastly, the summary output is shown after the 'OK' button is pressed.

2.4 Test Hypotheses

In order to determine whether the fitted line follows linear relationship, test hypotheses is done by comparing the value of F in ANOVA with fα,1,n-2. The value of F in ANOVA can be obtained directly from the regression summary output. The appropriate hypotheses are H0: β1=0 (there is no linear relationship between X and Y) and H1: β1≠0(there is linear relationship between X and Y). Thus, H0 is rejected if F> fα,1,n-2 and Ho is failed to reject if F< fα,1,n-2. When H0 is rejected, it means that the dependent and independent variable has linear relationship.

2.5 Confidence Interval

In order to find the 95% confidence interval for the population mean when the when the freshwater growth is 100, the formula of confidence interval about the mean response is used. The unknown values are obtained from regression summary output and applying of equation as shown below in order to obtain the results.

5.0 Results

5.1 Results for All Fishes:

Freshwater Growth

Marine Growth

147

131

444

405

139

113

446

422

160

137

438

428

99

121

437

469

120

139

405

424

151

144

435

402

115

161

394

440

121

107

406

410

109

129

440

366

119

123

414

422

130

148

444

410

110

129

465

352

127

119

457

414

100

134

498

396

115

139

452

473

117

140

418

398

112

126

502

434

116

116

478

395

98

112

500

334

98

117

589

455

83

97

480

439

85

134

424

511

88

88

455

432

98

99

439

381

74

105

423

418

58

112

411

475

114

98

484

436

88

80

447

431

77

139

448

515

86

97

450

508

86

103

493

429

65

93

495

420

127

85

470

424

91

60

454

456

76

115

430

491

44

113

448

474

42

91

512

421

50

109

417

451

57

122

466

442

42

Table 5.1: Data for All Fishes

68

496

363

SUMMARY OUTPUT:

Regression Statistics

Multiple R

0.172822174

R Square

0.029867504

Adjusted R Square

0.017429908

Standard Error

40.7184312

Observations

80

ANOVA

 

df

SS

MS

F

Significance F

Regression

1

3981.480125

3981.480125

2.401388784

0.125275736

Residual

78

129323.2699

1657.990639

Total

79

133304.75

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Intercept

469.6160571

18.31507653

25.64095521

9.42156E-40

FreshM

-0.257920086

0.166438551

-1.549641502

0.125275736

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

433.1535413

506.078573

433.1535413

506.078573

-0.589273781

0.07343361

-0.58927378

0.07343361

Figure 5.1: Scatter Diagram & Fitted Line of All Fish

RESIDUAL OUTPUT:

Observation

Predicted Marine Growth

Residuals

1

431.7018

12.2982

2

433.7652

12.23483

3

428.3488

9.651157

4

444.082

-7.08197

5

438.6656

-33.6656

6

430.6701

4.329876

7

439.9552

-45.9552

8

438.4077

-32.4077

9

441.5028

-1.50277

10

438.9236

-24.9236

11

436.0864

7.913554

12

441.2448

23.75515

13

436.8602

20.13979

14

443.824

54.17595

15

439.9552

12.04475

16

439.4394

-21.4394

17

440.729

61.27099

18

439.6973

38.30267

19

444.3399

55.66011

20

444.3399

144.6601

21

448.2087

31.79131

22

447.6928

-23.6928

23

446.9191

8.08091

24

444.3399

-5.33989

25

450.53

-27.53

26

454.6567

-43.6567

27

440.2132

43.78683

28

446.9191

0.08091

29

449.7562

-1.75621

30

447.4349

2.56507

31

447.4349

45.56507

32

452.8513

42.14875

33

436.8602

33.13979

34

446.1453

7.854671

35

450.0141

-20.0141

36

458.2676

-10.2676

37

458.7834

53.21659

38

456.7201

-39.7201

39

454.9146

11.08539

40

458.7834

37.21659

41

435.8285

-30.8285

42

440.4711

-18.4711

43

434.281

-6.28101

44

438.4077

30.59227

45

433.7652

-9.76517

46

432.4756

-30.4756

47

428.0909

11.90908

48

442.0186

-32.0186

49

436.3444

-70.3444

50

437.8919

-15.8919

51

431.4439

-21.4439

52

436.3444

-84.3444

53

438.9236

-24.9236

54

435.0548

-39.0548

55

433.7652

39.23483

56

433.5072

-35.5072

57

437.1181

-3.11813

58

439.6973

-44.6973

59

440.729

-106.729

60

439.4394

15.56059

61

444.5978

-5.59781

62

435.0548

75.94523

63

446.9191

-14.9191

64

444.082

-63.082

65

442.5344

-24.5344

66

440.729

34.27099

67

444.3399

-8.33989

68

448.9825

-17.9825

69

433.7652

81.23483

70

444.5978

63.40219

71

443.0503

-14.0503

72

445.6295

-25.6295

73

447.6928

-23.6928

74

454.1409

1.859148

75

439.9552

51.04475

76

440.4711

33.52891

77

446.1453

-25.1453

78

441.5028

9.497232

79

438.1498

3.850193

80

452.0775

-89.0775

5.2 Results for Males' Fish:

Freshwater Growth

Marine Growth

147

83

444

480

139

85

446

424

160

88

438

455

99

98

437

439

120

74

405

423

151

58

435

411

115

114

394

484

121

88

406

447

109

77

440

448

119

86

414

450

130

86

444

493

110

65

465

495

127

127

457

470

100

91

498

454

115

76

452

430

117

44

418

448

112

42

502

512

116

50

478

417

98

57

500

466

98

42

589

496

Table 5.2: Data for Males Fish

SUMMARY OUTPUT:

Regression Statistics

Multiple R

0.190576376

R Square

0.036319355

Adjusted R Square

0.010959338

Standard Error

37.05170732

Observations

40

ANOVA

 

df

SS

MS

F

Significance F

Regression

1

1966.097431

1966.097431

1.43215026

0.238826934

Residual

38

52167.50257

1372.829015

Total

39

54133.6

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Intercept

478.3539243

20.29522918

23.569772

2.77485E-24

Freshwater Growth

-0.236440512

0.197573001

-1.196724806

0.238826934

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

437.2683812

519.4394675

437.2683812

519.4394675

-0.636406139

0.163525116

-0.636406139

0.163525116

Figure 5.2: Scatter Diagram & Fitted Line of Males Fish

RESIDUAL OUTPUT:

Observation

Predicted Marine Growth

Residuals

1

443.5971691

0.402830894

2

445.4886932

0.511306801

3

440.5234425

-2.523442454

4

454.9463137

-17.94631367

5

449.9810629

-44.98106292

6

442.6514071

-7.651407059

7

451.1632655

-57.16326548

8

449.7446224

-43.74462241

9

452.5819086

-12.58190855

10

450.2175034

-36.21750343

11

447.6166578

-3.616657805

12

452.345468

12.65453196

13

448.3259793

8.67402066

14

454.7098732

43.29012684

15

451.1632655

0.83673452

16

450.6903845

-32.69038446

17

451.872587

50.12741298

18

450.926825

27.07317503

19

455.1827542

44.81724582

20

455.1827542

133.8172458

21

458.7293619

21.27063815

22

458.2564808

-34.25648083

23

457.5471593

-2.547159296

24

455.1827542

-16.18275418

25

460.8573265

-37.85732646

26

464.6403746

-53.64037465

27

451.399706

32.60029401

28

457.5471593

-10.5471593

29

460.1480049

-12.14800492

30

458.0200403

-8.020040319

31

458.0200403

34.97995968

32

462.9852911

32.01470893

33

448.3259793

21.67402066

34

456.8378378

-2.837837761

35

460.3844454

-30.38444544

36

467.9505418

-19.95054181

37

468.4234228

43.57657717

38

466.5318987

-49.53189874

39

464.8768152

1.123184841

40

468.4234228

27.57657717

5.3 Results for Females' Fish:

Freshwater growth

Marine growth

131

97

405

439

113

134

422

511

137

88

428

432

121

99

469

381

139

105

424

418

144

112

402

475

161

98

440

436

107

80

410

431

129

139

366

515

123

97

422

508

148

103

410

429

129

93

352

420

119

85

414

424

134

60

396

456

139

115

473

491

140

113

398

474

126

91

434

421

116

109

395

451

112

122

334

442

117

68

455

363

Table 5.3: Data for Females Fish

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.040178762

R Square

0.001614333

Adjusted R Square

-0.024658974

Standard Error

41.54801703

Observations

40

ANOVA

 

df

SS

MS

F

Significance F

Regression

1

106.0666754

106.0666754

0.061443841

0.805562865

Residual

38

65597.03332

1726.237719

Total

39

65703.1

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Intercept

420.6274808

35.00378835

12.01662736

1.63513E-14

Freshwater Growth

0.074221808

0.299427962

0.247878681

0.805562865

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

349.7660166

491.4889451

349.7660166

491.4889451

-0.531938406

0.680382023

-0.531938406

0.680382023

Figure 5.3: Scatter Diagram & Fitted Line of Females Fish

RESIDUAL OUTPUT:

Observation

Predicted Marine Growth

Residuals

1

430.3505378

-25.35053775

2

429.0145452

-7.0145452

3

430.7958686

-2.795868602

4

429.6083197

39.39168033

5

430.9443122

-6.944312219

6

431.3154213

-29.31542126

7

432.577192

7.422807995

8

428.5692143

-18.56921435

9

430.2020941

-64.20209413

10

429.7567633

-7.756763284

11

431.6123085

-21.6123085

12

430.2020941

-78.20209413

13

429.4598761

-15.45987605

14

430.5732032

-34.57320318

15

430.9443122

42.05568778

16

431.018534

-33.01853403

17

429.9794287

4.020571291

18

429.2372106

-34.23721062

19

428.9403234

-94.94032339

20

429.3114324

25.68856757

21

427.8269963

11.17300374

22

430.5732032

80.42679682

23

427.159

4.841000012

24

427.9754399

-46.97543988

25

428.4207707

-10.42077073

26

428.9403234

46.05967661

27

427.9012181

8.098781927

28

426.5652255

4.434774479

29

430.9443122

84.05568778

30

427.8269963

80.17300374

31

428.2723271

0.727672885

32

427.530109

-7.530109031

33

426.9363346

-2.936334563

34

425.0807894

30.91921065

35

429.1629888

61.83701118

36

429.0145452

44.9854548

37

427.3816654

-6.381665414

38

428.717658

22.28234203

39

429.6825415

12.31745852

40

425.6745638

-62.67456382

6.0 Discussion and Analysis

6.1 Discussion and analysis for All Fishes:

Figure 5.1 shows the scatter diagram and fitted line of all fishes, while the summary output of all fish calculated using Microsoft Excel(Data Analysis) also shown in section 5.1. In Figure 5.1, we noticed that there is equation of straight line and its respective coefficient of determination, R2. The value from the equation can be obtained from the summary output as well, which is the coefficients of intercept and freshwater growth. The straight line equation is y = -0.2579x + 469.62 and the value of R2 = 0.0299.

In order to determine if β1 differs from zero, a test hypothesis is done where,

H0: β1 = 0

H1: β1 ≠ 0

From summary output, the F in ANOVA table is 2.4014. Thus, we reject H0 if F > f0.05,1,78 ≈ 4(Value from booklet). Since F = 2.4014 < f0.05,1,78 ≈ 4, so we fail to reject H0. Therefore there is no strong evidence to determine that β1 differs from zero.

To find 95% confidence interval for the population mean when the freshwater growth is 100. Thus, the freshwater growth (x0) is 100, therefore

Where = 469.62 and = -0.2579. Substitute these values and = 443.83.

Then, by using the formula

Where t0.025,78 ≈ 2.00, = 1657.99, n = 80, = 106.5875, = 59851.3875. All values are substituted into the equation and thus the 95% confidence interval for the population mean is

434.46 ≤ ≤ 453.20

6.2 Discussion and analysis for Males Fish:

Figure 5.2 shows the scatter diagram and fitted line of males, while the summary output of males fish calculated using Microsoft Excel(Data Analysis) also shown in section 5.2. In Figure 5.2, we noticed that there is equation of straight line and its respective coefficient of determination, R2. The value from the equation can be obtained from the summary output as well, which the coefficient of intercept = β0 while the coefficient of freshwater growth = β1.The straight line equation is y = -0.2364x + 478.35 and R2 = 0.0363

Test hypothesis is done in order to determine if β1 differs from zero. Therefore,

H0: β1 = 0

H1: β1 ≠ 0

From the summary output, the F in ANOVA table is 1.4322. Thus, we reject H0 if F > f0.05,1,38 ≈ 4.08. Since F = 1.4322 < f0.05,1,78 ≈ 4.08, so fail to reject H0. Therefore there is no strong evidence to determine that β1 differs from zero.

To obtain the 95% confidence interval for the population mean when the freshwater growth (x0) is 100, by applying formula

Where = 478.35 and = -0.2364. Substitute these values and = 454.71.

Then, by using the formula

Where t0.025,38 ≈ 2.021, = 1372.83, n = 40, = 98.35, = 35169.1. All values are substituted into the equation and thus the is within the confidence interval of

442.85 ≤ ≤ 466.57

6.3 Discussion and analysis for Females Fish:

Figure 5.3 shows the scatter diagram and fitted line of males, while the summary output of males fish calculated using Microsoft Excel(Data Analysis) also shown in section 5.3. In Figure 5.3, we noticed that there is equation of straight line and its respective coefficient of determination, R2. The value from the equation can be obtained from the summary output as well, which the coefficient of intercept = β0 while the coefficient of freshwater growth = β1.The straight line equation is y = 0.0742x + 420.63 and R2 = 0.0016

Test hypothesis is done in order to determine if β1 differs from zero. Therefore,

H0: β1 = 0

H1: β1 ≠ 0

From the summary output, the F in ANOVA table is 0.0614. Thus, we reject H0 if F > f0.05,1,38 ≈ 4.08. Since F = 0.0614 < f0.05,1,78 ≈ 4.08, so we fail to reject H0. Therefore there is no strong evidence to determine that β1 differs from zero.

The 95% confidence interval for the population mean when the freshwater growth is 100 is found by following these steps and by using formula.

Where = 420.63 and = 0.0742. Substitute these values and = 428.05.

Then, by using the formula

Where t0.025,38 ≈ 2.021, = 1726.24, n = 40, = 114.825, = 19253.775. All values are substituted into the equation and thus the is within the confidence interval of

412.03 ≤ ≤ 444.07

7.0 Conclusion

In conclusion, regression analysis can be used to model the relationship between one or more response variables and one or more predictor variables. Based on the task given, it is a simple linear regression analysis which consists of a single predictor, or regressor which is the salmon's growth in fresh water and a single response variable which is the salmon's growth in marine. This simple linear regression analysis determines whether there is any relationship between the regressor and the response variable. The simple linear regression model also gives a straight line relationship between a single response (dependent) variable and a single predictor (independent) variable. In this task regression analysis have been done separately for all fishes, males' fish and females' fish.

From the results obtained, the marine growth on freshwater growth for all fishes can be represented by a straight line equation: . We fail to reject H0 since there is no sufficient evidence to determine that β1 differs from zero. This shows that there is no linear relationship between the Y and X. The 95% confidence interval for the population mean when the freshwater is 100 is between 434.46 and 453.20.

On the other hand, for the males' fish, is the straight line equation obtained from the scatter diagram and fitted line plot. We fail to reject H0 and thus we can statistically conclude that there is no linear relationship between Y and X. The 95% confidence interval for the population mean when the freshwater is 100 is between of 442.85 and 466.57.

Besides, for females' fish, the straight line equation obtained is . Since the H0 is failed to reject, therefore we can statistically conclude that there is no linear relationship between Y and X. The 95% confidence interval for the population mean when the freshwater is 100 is 412.03 ≤ ≤ 444.07.

Lastly, from the results and analysis obtained, it is very obvious and we can therefore statically conclude that the fresh water growth and marine growth salmons do not have direct relationship. This is because the data obtained do not provide strong evidence to show the existence of linearity between fresh water growth and marine growth.

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