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Answer :
To calculate the correlation coefficient, which measures the strength and direction of a linear relationship between two variables, we'll go through the following steps:
1. List the Data:
- We have two lists of values: `x` and `y`.
- [tex]\( x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] \)[/tex]
- [tex]\( y = [44, 40.65, 37.9, 36.05, 32.7, 29.75, 26.6, 24.75, 25, 21.55, 19.8, 16.15, 15, 8.55, 9.5] \)[/tex]
2. Calculate the Means:
- First, find the mean (average) of each list.
- [tex]\( \text{Mean of } x (\bar{x}) = \frac{1+2+3+\dots+15}{15} = 8 \)[/tex]
- [tex]\( \text{Mean of } y (\bar{y}) = \frac{44+40.65+37.9+\dots+9.5}{15} \approx 25.43 \)[/tex]
3. Find the Deviations:
- Calculate the deviations of each value from their respective means:
- [tex]\( \left(x_i - \bar{x}\right) \)[/tex] for each [tex]\( x_i \)[/tex] in [tex]\( x \)[/tex]
- [tex]\( \left(y_i - \bar{y}\right) \)[/tex] for each [tex]\( y_i \)[/tex] in [tex]\( y \)[/tex]
4. Calculate the Covariance:
- Sum the product of the deviations for x and y:
- Compute [tex]\( \sum (x_i - \bar{x})(y_i - \bar{y}) \)[/tex]
5. Calculate the Variances:
- Calculate the variance for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- [tex]\( \sum (x_i - \bar{x})^2 \)[/tex]
- [tex]\( \sum (y_i - \bar{y})^2 \)[/tex]
6. Calculate the Correlation Coefficient [tex]\( r \)[/tex]:
- Use the formula:
[tex]\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}
\][/tex]
7. Round the Result:
- The correlation coefficient is calculated to be approximately [tex]\(-0.994\)[/tex].
The correlation coefficient, [tex]\( r = -0.994 \)[/tex], suggests a strong negative linear relationship between the data sets [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This means that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] tends to decrease in a nearly linear manner.
1. List the Data:
- We have two lists of values: `x` and `y`.
- [tex]\( x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] \)[/tex]
- [tex]\( y = [44, 40.65, 37.9, 36.05, 32.7, 29.75, 26.6, 24.75, 25, 21.55, 19.8, 16.15, 15, 8.55, 9.5] \)[/tex]
2. Calculate the Means:
- First, find the mean (average) of each list.
- [tex]\( \text{Mean of } x (\bar{x}) = \frac{1+2+3+\dots+15}{15} = 8 \)[/tex]
- [tex]\( \text{Mean of } y (\bar{y}) = \frac{44+40.65+37.9+\dots+9.5}{15} \approx 25.43 \)[/tex]
3. Find the Deviations:
- Calculate the deviations of each value from their respective means:
- [tex]\( \left(x_i - \bar{x}\right) \)[/tex] for each [tex]\( x_i \)[/tex] in [tex]\( x \)[/tex]
- [tex]\( \left(y_i - \bar{y}\right) \)[/tex] for each [tex]\( y_i \)[/tex] in [tex]\( y \)[/tex]
4. Calculate the Covariance:
- Sum the product of the deviations for x and y:
- Compute [tex]\( \sum (x_i - \bar{x})(y_i - \bar{y}) \)[/tex]
5. Calculate the Variances:
- Calculate the variance for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- [tex]\( \sum (x_i - \bar{x})^2 \)[/tex]
- [tex]\( \sum (y_i - \bar{y})^2 \)[/tex]
6. Calculate the Correlation Coefficient [tex]\( r \)[/tex]:
- Use the formula:
[tex]\[
r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}
\][/tex]
7. Round the Result:
- The correlation coefficient is calculated to be approximately [tex]\(-0.994\)[/tex].
The correlation coefficient, [tex]\( r = -0.994 \)[/tex], suggests a strong negative linear relationship between the data sets [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This means that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] tends to decrease in a nearly linear manner.
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