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1. Standard Error of Estimate:

The standard error of estimation is an estimated standard deviation of the error term u. It also known as standard error of the regression. Standard Error of Estimate is shows variation of observations. It is applied to inspect the accuracy of estimation made. Standard error of estimate tells the accuracy of the estimated figures.

Formula of the standard error of estimate: sqrt(SSE/(n-k))

Following is the calculation of Standard Error of Estimate:

SSE

25843.41

k

2

n

400

N-k

398

SSE/(n-k)

64.9331909548

sqrt(SSE/(n-k))

8.058113362

If standard error is small, the data will be more representative of the true mean. In cases where the standard error is large, the data may have some notable irregularities.

2. Coefficient of Determination:

The coefficient of determination is a statistical analysis that determine explanation of the model and estimated future outcomes. It shows the level of related variability in the data set. The coefficient of determination refers to R-squared and applied to determine correctness of the model. Coefficient of determination tells that variables in given model is certain percentage of observed variation. It is represented as a value between 0 and 1. Closer the value is to 1, the better the fit, or relationship, between the two factors. Thus, in case the R Square is equal to 0.2672, then approximately less than half of the observed variation can be explained by the model.

Formula of Coefficient of determination: MSS/TSS = (TSS − RSS)/TSS

Where MSS is the model sum of squares, RSS is the residual sum of squares and TSS is the total sum of squares associated with the outcome variable.

3. The Adjusted Coefficient of Determination for a Degree of Freedom:

Adjusted coefficient of determination is best for a model with several variables, such as a multiple regression model. Adjusted R-squared provides percentage of variation interpreted by only those independent variables that in reality affect the dependent variable.

Following is the calculation of the Adjusted coefficient of determination:


Adjusted coefficient of determination =

 

 
     
     
 

1-(1-0.2672)[(400-1)/400-(2+1)]

 

0.2635083123

 

4.Overall Utility Level of Model:

ANOVA table basis of calculation:

 

df

SS

MS

F

 

Regression

k

SSR

MSR = SSR/k

F=MSR/MSE

 

Residual

n-k-1

SSE

MSE= SSE/(n-k-1)

 

 

Total

n-1

SST

   

 

ANOVA

 

df

SS

MS

F

Significance F

Regression

2

9421.58

4710.79

72.37

0

Residual

397

25843.41

65.1

 

 

Total

399

35264.98

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper

95%

Intercept

93.8993

8.0072

11.7269

0

78.1575

109.641

X1

0.4849

0.0412

11.7772

0

0.404

0.5659

X2

-0.0229

0.0395

-0.5811

0.56

-0.1005

0.0546

From the above Excel outputs, the value of the test statistic for testing the overall utility of the model, F = 72.37, the out put also includes the P-value of the test, which is 0.00

As p-value = 0.00 < 0.05 = alpha, hence this model is useful at 5% level of significance.

5. Interpretation of the Coefficients:

Regression coefficients exhibits fluctuation in mean in the response variable for one unit of change in predictor variable while holding other predictors in the model constant. This is important because it differentiates the role of one variable from all of the others in the model.

Coefficients helps to determine whether there is a positive or negative correlation between each independent or dependent variable. A positive coefficient indicates that with increase in the value of the independent variable, mean of the dependent variable also leads to increase. A negative coefficient suggests that with the increase in independent variable increases, the dependent variable leads to decrease. Coefficient value shows the extent to which the mean of the dependent variable changes given a one-unit shift in the independent variable while holding other variables in the model constant.

6. Relationship between Heights of Sons and Fathers:

The regression line for showing a relationship between heights of the sons and the fathers is:

y = b0+ b1x

y = 93.8993 + 0.4849x

If there is no linear relationship between these variables, then b1=0. If there is a linear relationship, then b1≠ 0 hence these data allow the statistic practitioner to infer that the heights of sons and fathers are linearly related.

7. Relationship Between Heights of Sons and Mother:

Regression line for showing relationship between heights of the sons and the fathers is:

y = b0+ b1X

y = 93.8993 + (-0.0229) x

If there is no linear relationship between these variables, b1=0. If there is a linear relationship, then ≠ 0 hence these data allow the statistic practitioner to infer that the heights of sons and mothers are linearly related but they is a negative correlation.

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