Respuesta :
Answer:
a)X: 0 2 6 10 13 15 18
Y:263 268 271 272 276 277 279
X represent the number of years since 1990
n=7 [tex] \sum x = 64, \sum y = 1906, \sum xy= 17647, \sum x^2 =858, \sum y^2 =519164[/tex]
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]
[tex]r=\frac{7(17647)-(64)(1906)}{\sqrt{[7(858) -(64)^2][7(519164) -(1906)^2]}}=0.97599[/tex]
[tex]\bar X = \frac{\sum_{i=1^n X_i}}{n} = 9.14286[/tex]
[tex]\bar Y = \frac{\sum_{i=1^n Y_i}}{n} = 272.286[/tex]
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=858-\frac{64^2}{7}=272.857[/tex]
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}=17647-\frac{64*1906}{7}=220.714[/tex]
And the slope would be:
[tex]m=\frac{220.714}{272.857}=0.809[/tex]
Now we can find the means for x and y like this:
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=272.286-(0.809*9.143)=264.889[/tex]
So the line would be given by:
[tex]y=0.809 x +264.889[/tex]
b) For this case the percent of variation in scores is explained by the linear trend is given by the determination coefficient [tex] r^2[/tex] and we got:
[tex] r^2 =0.976^2 = 0.9526[/tex]
So then we can say that the percent of variation explained is approximately 95.26%
Step-by-step explanation:
Pearson correlation coefficient(r), "measures a linear dependence between two variables (x and y). Its a parametric correlation test because it depends to the distribution of the data. And other assumption is that the variables x and y needs to follow a normal distribution".
Solution to the problem
Part a
We assume the following data:
X: 0 2 6 10 13 15 18
Y:263 268 271 272 276 277 279
X represent the number of years since 1990
n=7 [tex] \sum x = 64, \sum y = 1906, \sum xy= 17647, \sum x^2 =858, \sum y^2 =519164[/tex]
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]
[tex]r=\frac{7(17647)-(64)(1906)}{\sqrt{[7(858) -(64)^2][7(519164) -(1906)^2]}}=0.97599[/tex]
So then the correlation coefficient would be r =0.976
The mean for X on this case is given by:
[tex]\bar X = \frac{\sum_{i=1^n X_i}}{n} = 9.14286[/tex]
[tex]\bar Y = \frac{\sum_{i=1^n Y_i}}{n} = 272.286[/tex]
For this case we need to calculate the slope with the following formula:
[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]
Where:
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]
So we can find the sums like this:
[tex]\sum_{i=1}^n x_i = 64[/tex]
[tex]\sum_{i=1}^n y_i =1906[/tex]
[tex]\sum_{i=1}^n x^2_i =858[/tex]
[tex]\sum_{i=1}^n y^2_i =519164[/tex]
[tex]\sum_{i=1}^n x_i y_i =17647[/tex]
With these we can find the sums:
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=858-\frac{64^2}{7}=272.857[/tex]
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}=17647-\frac{64*1906}{7}=220.714[/tex]
And the slope would be:
[tex]m=\frac{220.714}{272.857}=0.809[/tex]
Now we can find the means for x and y like this:
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=272.286-(0.809*9.143)=264.889[/tex]
So the line would be given by:
[tex]y=0.809 x +264.889[/tex]
Part b
For this case the percent of variation in scores is explained by the linear trend is given by the determination coefficient [tex] r^2[/tex] and we got:
[tex] r^2 =0.976^2 = 0.9526[/tex]
So then we can say that the percent of variation explained is approximately 95.26%
