Answer:
a) n=5 [tex] \sum x = 30, \sum y = 25, \sum xy = 162, \sum x^2 =206, \sum y^2 =155[/tex]
[tex]r=\frac{5(162)-(30)(25)}{\sqrt{[5(206) -(30)^2][5(155) -(25)^2]}}=0.42967[/tex]
So then the correlation coefficient would be r =0.430 rounded
b) Null hypothesis: [tex]\rho =0[/tex]
Alternative hypothesis: [tex]\rho \neq 0[/tex]
Because the correlation coeffcient is positive and the absolute value of the correlation coefficient 0.430 is not greater than the critical value for this dataset, no linear relation exists between x and y.
And the reason is because we fail to reject the null hypothesis.
Step-by-step explanation:
Part a
We have the following data:
x: 2 6 6 7 9
y: 3 2 6 9 5
The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1.
And in order to calculate the correlation coefficient we can use this formula:
[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]
For our case we have this:
n=5 [tex] \sum x = 30, \sum y = 25, \sum xy = 162, \sum x^2 =206, \sum y^2 =155[/tex]
[tex]r=\frac{5(162)-(30)(25)}{\sqrt{[5(206) -(30)^2][5(155) -(25)^2]}}=0.42967[/tex]
So then the correlation coefficient would be r =0.430 rounded
Part b
In order to test the hypothesis if the correlation coefficient it's significant we have the following hypothesis:
Null hypothesis: [tex]\rho =0[/tex]
Alternative hypothesis: [tex]\rho \neq 0[/tex]
The statistic to check the hypothesis is given by:
[tex]t=\frac{r \sqrt{n-2}}{\sqrt{1-r^2}}[/tex]
And is distributed with n-2 degreed of freedom. df=n-2=5-2=3
In our case the value for the statistic would be:
[tex]t=\frac{0.430\sqrt{5-2}}{\sqrt{1-(0.430)^2}}=0.825[/tex]
The critical value for n =5 is given by the table attached. We can see that the critical value is [tex] r_{crit}= 0.878[/tex], and then the final conclusion would be:
Because the correlation coeffcient is positive and the absolute value of the correlation coefficient 0.430 is not greater than the critical value for this dataset, no linear relation exists between x and y
And the reason is because we fail to reject the null hypothesis.
