Questions: correlation using the level r>0.9. - Σx=40 - (Σx)^2=1600 - Σy=50 - (Σy)^2=2500 - Σxy=600 - Σx^2=470 - Σy^2=750 - n=4 r=(n(Σxy)-(Σx)(Σy))/sqrt([n Σx^2-(Σx)^2][n Σy^2-(Σy)^2]) A. r= Round this to two decimals. B. The strength is (type strong or weak) and the direction is (type positive or negative)

correlation using the level r>0.9.
- Σx=40
- (Σx)^2=1600
- Σy=50
- (Σy)^2=2500
- Σxy=600
- Σx^2=470
- Σy^2=750
- n=4
r=(n(Σxy)-(Σx)(Σy))/sqrt([n Σx^2-(Σx)^2][n Σy^2-(Σy)^2])
A. r= Round this to two decimals.
B. The strength is (type strong or weak) and the direction is (type positive or negative)

Transcript text: correlation using the level $|\mathrm{r}|>0.9$. - $\Sigma x=40$ - $(\Sigma x)^{2}=1600$ - $\Sigma y=50$ - $(\Sigma y)^{2}=2500$ - $\Sigma x y=600$ - $\Sigma x^{2}=470$ - $\Sigma y^{2}=750$ - $n=4$ $r=\frac{n(\Sigma x y)-(\Sigma x)(\Sigma y)}{\sqrt{\left[n \sum x^{2}-\left(\sum x\right)^{2}\right]\left[n \sum y^{2}-\left(\sum y\right)^{2}\right]}}$ A. $r=$ $\square$ Round this to two decimals. B. The strength is (type strong or weak) $\square$ and the direction is (type positive or negative) $\square$

Solution

Solution Steps

Step 1: Calculate the Correlation Coefficient

Using the provided data, we calculate the correlation coefficient $ r $ using the formula:

\[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{\left[n \Sigma x^2 - (\Sigma x)^2\right]\left[n \Sigma y^2 - (\Sigma y)^2\right]}} \]

Substituting the values:

$ n = 4 $
$ \Sigma x = 40 $
$ \Sigma y = 50 $
$ \Sigma xy = 600 $
$ \Sigma x^2 = 470 $
$ \Sigma y^2 = 750 $

The numerator is calculated as:

\[ n(\Sigma xy) - (\Sigma x)(\Sigma y) = 4(600) - (40)(50) = 2400 - 2000 = 400 \]

The denominator is calculated as:

\[ \sqrt{\left[4(470) - (40)^2\right]\left[4(750) - (50)^2\right]} = \sqrt{(1880 - 1600)(3000 - 2500)} = \sqrt{280 \cdot 500} = \sqrt{140000} \approx 374.165 \]

Thus, the correlation coefficient $ r $ is:

\[ r = \frac{400}{374.165} \approx 1.07 \]

Step 2: Round the Correlation Coefficient

Rounding $ r $ to two decimal places gives:

\[ r \approx 1.07 \]

Step 3: Determine Strength and Direction

To assess the strength and direction of the correlation:

The strength is classified as "strong" if $ |r| > 0.9 $. Since $ |1.07| > 0.9 $, the strength is "strong".
The direction is "positive" since $ r > 0 $.