Question

Given the following data set X3 4 5 7 8 7 4 6 7 12 14 a) Draw a Scaker diagram b) Find the coefficient of of determination and Correlation c R² and coefficient c Determine If it is if linear the relation determine is the linear or nonlineare linear regression line d) If line in the and (a) relation sketch above. is linear on the determine the regression same Cordinate System

Answer 1

a) Scatter Diagram is

Scatter Plot 0 1 2 3 4 5 6 7 8 9

b) For finding the correlation coefficient
we find the means of X and Y using the excel function Average
Mean of X = X̅ = 5.4
Mean of Y = Y̅ = 8.6

X	Y	x2	y2	xy	(xi-x̅)	(yi-y̅)	(xi-x̅)2	(yi-y̅)2	(xi-x̅)(yi-y̅)
3	4	9.00	16.00	12.00	-2.40	-4.60	5.76	21.16	11.04
4	6	16.00	36.00	24.00	-1.40	-2.60	1.96	6.76	3.64
5	7	25.00	49.00	35.00	-0.40	-1.60	0.16	2.56	0.64
7	12	49.00	144.00	84.00	1.60	3.40	2.56	11.56	5.44
8	14	64.00	196.00	112.00	2.60	5.40	6.76	29.16	14.04
						Total	17.2	71.2	34.8

Correlation Coefficient r is calculated as  
We calculate correlation coefficient  
n = number of observations = 5
Mean of X = X̅ = 5.4
Mean of Y = Y̅ = 8.6

Σ ας – ) ( – y) Σ=1(α) - )?Σ:10 - y)2 34.8 17.2 * 71.2 γ = 0.9944

Correlation Coefficient r = 0.9944  
  
Coefficient of Determination  
R² = (0.9944)²  
R² = 0.9888  

c) From the scatter plot, we can see that the points are mostly on a straight line   
Hence, the relation between X and Y is linear   
Using the Excel tool for Regression at the option Data->Data Analysis->Regression, we find the slope and the intercept   

	Coefficients
Intercept	-2.3256
X	2.0233

The regression line is
ŷ = 0.0233 x - 2.3256

d) From the regression equation , we get Predicted Y for each X value

X	Y	Predicted Y
3	4	3.74
4	6	5.77
5	7	7.79
7	12	11.84
8	14	13.86

Plotting the regression line (Predicted Y) on the same scatter plot  

Regression Line Predicted Y 1 2 3 4 5 6 7 8 X