Respuesta :
Answer:
[tex]t=\frac{-0.9690\sqrt{10-2}}{\sqrt{1-(-0.9690)^2}}=-11.107[/tex]
Step-by-step explanation:
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=10 [tex] \sum x = 103, \sum y = 47, \sum xy = 343, \sum x^2 =1347, \sum y^2 =295[/tex]
[tex]r=\frac{10(343)-(103)(47)}{\sqrt{[10(1347) -(103)^2][10(295) -(47)^2]}}=-0.9691[/tex]
So then the correlation coefficient would be r =-0.9690
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=10-2=8
In our case the value for the statistic would be:
[tex]t=\frac{-0.9690\sqrt{10-2}}{\sqrt{1-(-0.9690)^2}}=-11.107[/tex]
Since [tex]\alpha=0.01[/tex] then [tex]\alpha/2 =0.005[/tex] and using the degrees of freedom we see that the critical values that accumulates 0.005 of the area on each tail are:
[tex]t_{\alpha/2}=-3.36[/tex] and [tex]t_{1-\alpha/2}=3.36[/tex]
And in our case since our calculated value it's outside of the interval (-3.36;3.36) we can reject the null hypothesis at 1% the significance.
