Use the following information regarding a variable Y and a v

Use the following information regarding a variable (Y) and a variable (X) to answer this next 6 questions. n = 8 x = 56 y = 296 x 2 = 428 y 2 = 11,920 xy = 2,257.

Find sample covariance, sample correlation coefficient, direct relation, and calculated value of the test statistic and critical value.

Solution

a & b)
r( X,Y) =    Co V ( X,Y) / S.D (X) * S.D (y)                      
r( X,Y) =    Sum(XY) / N- Mean of (X) * Mean of (Y) / Sqrt( X^2/n - ( Mean of X)^2 ) Sqrt( Y^2/n - ( Mean of Y)^2 )                        
                          
Co v ( X, Y ) =   1 /8 (2257) - [ 1/8 *56 ] [ 1/8 *296] = 23.125          
S. D ( X ) =   Sqrt( 1/8*428-(1/8*56)^2) = 2.121          
S .D (Y) =    Sqrt( 1/8*11920-(1/8*296)^2) = 11          
r(x,y) =    23.125 / 2.121*11   =   0.9912              
                          
If r = 0.9912> 0 ,Perfect Positive Correlation                          

c)
Test For correlation co-efficient
Set Up Hypothesis
Under The Null Hypothesis H0: =0
Under The Alternate Hypothesis H1: !=0
Test Statistic
Value of ( r ) =0.9912
Number (n)=8
we use Test Statistic (t) = r / Sqrt(1-r^2/(n-2))
to=0.9912/(Sqrt( ( 1-0.9912^2 )/(8-2) )
to =18.34
|to | =18.34
Critical Value
The Value of |t | at LOS 0.1% is 1.943
We got |to| =18.34 & | t | =1.943
Make Decision
Hence Value of | to | > | t | and Here we Reject Ho