Here's a step-by-step breakdown of the calculation of the correlation coefficient along with the data table:

Data Table:

Steps to Calculate Correlation Coefficient:

1. Sum the variables:

   • Sum of $x$: $\sum x = 410$
   • Sum of $y$: $\sum y = 281$
   • Sum of $x^2$: $\sum x^2 = 24050$
   • Sum of $y^2$: $\sum y^2 = 11355$
   • Sum of $xy$: $\sum xy = 16506$

2. Apply the Pearson Correlation formula: [ r = \frac{n \cdot \sum (xy) - \sum x \cdot \sum y}{\sqrt{[n \cdot \sum x^2 - (\sum x)^2] \cdot [n \cdot \sum y^2 - (\sum y)^2]}} ] Where:

   • $n = 7$ (number of data points)

   Substituting the values into the formula:

   Numerator: $7 \cdot 16506 - 410 \cdot 281 = 115542 - 115210 = 332$

   Denominator: [ \sqrt{[7 \cdot 24050 - 410^2] \cdot [7 \cdot 11355 - 281^2]} = \sqrt{(168350 - 168100) \cdot (79485 - 78961)} = \sqrt{250 \cdot 524} \approx 362.09 ]

   Finally: $r = \frac{332}{362.09} \approx 0.917$

Correlation Coefficient:

$r \approx 0.917$

This indicates a strong positive correlation between the variables $x$ and $y$.

Do you want more details on any step or have any questions?

Additional Questions:

1. How is the regression equation derived from this data?
2. What does a correlation coefficient of 0.917 signify in practical terms?
3. How does one test the significance of this correlation coefficient?
4. What if there were more points added to the dataset—how would this impact the correlation?
5. Can the correlation be negative, and what would that imply for this type of data?

Tip:

When calculating the correlation coefficient, always ensure that both variables have a linear relationship, as the Pearson method assumes linearity between variables.