Answer :
To find the correlation coefficient between the yearly earnings ([tex]$x$[/tex]) and the corresponding amounts ([tex]$y$[/tex]), follow these steps:
1. Understand the Variables:
- [tex]$x$[/tex]: Yearly earnings in dollars.
- [tex]$y$[/tex]: Some corresponding financial amounts in dollars (in this example, they might represent savings, revenue, or another related financial metric).
2. Calculate the Mean of Each Variable:
- Calculate the mean (average) of the [tex]$x$[/tex] values.
- Calculate the mean (average) of the [tex]$y$[/tex] values.
3. Calculate the Differences from the Mean:
- For each [tex]$x$[/tex] value, subtract the mean of [tex]$x$[/tex] to find the deviation of [tex]$x$[/tex].
- For each [tex]$y$[/tex] value, subtract the mean of [tex]$y$[/tex] to find the deviation of [tex]$y$[/tex].
4. Compute the Products of Deviations:
- Multiply each pair of deviations (the deviation of [tex]$x$[/tex] and the corresponding deviation of [tex]$y$[/tex]).
5. Find the Sum of the Products:
- Sum up all the products of deviations calculated in the previous step.
6. Compute the Sum of Squares:
- For [tex]$x$[/tex], sum the squares of the deviations from its mean.
- For [tex]$y$[/tex], sum the squares of the deviations from its mean.
7. Calculate the Correlation Coefficient (r):
- Use the formula for Pearson's correlation coefficient:
[tex]\[
r = \frac{\sum \left( (x_i - \bar{x})(y_i - \bar{y}) \right)}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}}
\][/tex]
- [tex]$\sum \left( (x_i - \bar{x})(y_i - \bar{y}) \right)$[/tex] is the sum of products of deviations.
- [tex]$\sum (x_i - \bar{x})^2$[/tex] is the sum of square deviations for [tex]$x$[/tex].
- [tex]$\sum (y_i - \bar{y})^2$[/tex] is the sum of square deviations for [tex]$y$[/tex].
8. Interpret the Result:
- A correlation coefficient of about 0.918 indicates a strong positive linear relationship between yearly earnings and the corresponding amounts.
Therefore, the correlation coefficient is approximately 0.918, suggesting a strong positive relationship between the variables. This means that as the yearly earnings increase, the corresponding financial metric ([tex]$y$[/tex]) also tends to increase.
1. Understand the Variables:
- [tex]$x$[/tex]: Yearly earnings in dollars.
- [tex]$y$[/tex]: Some corresponding financial amounts in dollars (in this example, they might represent savings, revenue, or another related financial metric).
2. Calculate the Mean of Each Variable:
- Calculate the mean (average) of the [tex]$x$[/tex] values.
- Calculate the mean (average) of the [tex]$y$[/tex] values.
3. Calculate the Differences from the Mean:
- For each [tex]$x$[/tex] value, subtract the mean of [tex]$x$[/tex] to find the deviation of [tex]$x$[/tex].
- For each [tex]$y$[/tex] value, subtract the mean of [tex]$y$[/tex] to find the deviation of [tex]$y$[/tex].
4. Compute the Products of Deviations:
- Multiply each pair of deviations (the deviation of [tex]$x$[/tex] and the corresponding deviation of [tex]$y$[/tex]).
5. Find the Sum of the Products:
- Sum up all the products of deviations calculated in the previous step.
6. Compute the Sum of Squares:
- For [tex]$x$[/tex], sum the squares of the deviations from its mean.
- For [tex]$y$[/tex], sum the squares of the deviations from its mean.
7. Calculate the Correlation Coefficient (r):
- Use the formula for Pearson's correlation coefficient:
[tex]\[
r = \frac{\sum \left( (x_i - \bar{x})(y_i - \bar{y}) \right)}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}}
\][/tex]
- [tex]$\sum \left( (x_i - \bar{x})(y_i - \bar{y}) \right)$[/tex] is the sum of products of deviations.
- [tex]$\sum (x_i - \bar{x})^2$[/tex] is the sum of square deviations for [tex]$x$[/tex].
- [tex]$\sum (y_i - \bar{y})^2$[/tex] is the sum of square deviations for [tex]$y$[/tex].
8. Interpret the Result:
- A correlation coefficient of about 0.918 indicates a strong positive linear relationship between yearly earnings and the corresponding amounts.
Therefore, the correlation coefficient is approximately 0.918, suggesting a strong positive relationship between the variables. This means that as the yearly earnings increase, the corresponding financial metric ([tex]$y$[/tex]) also tends to increase.
