Understand the Problem
The image contains notes related to statistics and mathematics, specifically formulas for regression coefficients, covariance, means, variance, standard deviation, and correlation coefficient. It is an instructional resource for mathematical concepts rather than a specific question.
Answer
Use the formulas: - $ \bar{x} = \frac{\sum x_i}{n} $ - $ \text{cov}(x,y) = \frac{1}{n} \sum (x_i y_i) - \bar{x} \bar{y} $ - $ r = \frac{\text{Cov}(x,y)}{s_x s_y} $
Answer for screen readers
The formulas provided can be summarized and used as follows:
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Mean: $$ \bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n} $$
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Covariance: $$ \text{cov}(x,y) = \frac{1}{n} \sum (x_i y_i) - \bar{x} \bar{y} $$
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Variance: $$ s^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 $$
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Correlation Coefficient: $$ r = \frac{\text{Cov}(x,y)}{s_x s_y} $$
Steps to Solve
- Identify Key Formulas
First, recognize the relevant formulas provided in the notes:
- Mean: $$ \bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n} $$
- Covariance: $$ \text{cov}(x,y) = \frac{1}{n} \sum (x_i y_i) - \bar{x} \bar{y} $$
- Variance: $$ s_x^2 = \frac{1}{n} \sum (x_i - \bar{x})^2, \quad s_y^2 = \frac{1}{n} \sum (y_i - \bar{y})^2 $$
- Correlation coefficient: $$ r = \frac{\text{Cov}(x,y)}{s_x s_y} $$
- Calculate Mean/Average
Use the formulas to calculate the means:
- For ( x ): $$ \bar{x} = \frac{\sum x_i}{n} $$
- For ( y ): $$ \bar{y} = \frac{\sum y_i}{n} $$
- Calculate Covariance
Substitute the means into the covariance formula: $$ \text{cov}(x,y) = \frac{1}{n} \sum (x_i y_i) - \bar{x} \bar{y} $$
- Calculate Variance
Calculate the variance for both ( x ) and ( y ):
- For ( x ): $$ s_x^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 $$
- For ( y ): $$ s_y^2 = \frac{1}{n} \sum (y_i - \bar{y})^2 $$
- Calculate Correlation Coefficient
Finally, find the correlation coefficient using the covariance and standard deviations: $$ r = \frac{\text{Cov}(x,y)}{s_x s_y} $$
The formulas provided can be summarized and used as follows:
-
Mean: $$ \bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n} $$
-
Covariance: $$ \text{cov}(x,y) = \frac{1}{n} \sum (x_i y_i) - \bar{x} \bar{y} $$
-
Variance: $$ s^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 $$
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Correlation Coefficient: $$ r = \frac{\text{Cov}(x,y)}{s_x s_y} $$
More Information
These formulas are fundamental in statistics for understanding relationships between variables. They can be applied to analyze data sets in various fields, such as economics, psychology, and natural sciences.
Tips
- Neglecting Zero Values: Forgetting to account for zeros in data sets can skew results.
- Using Incorrect Formulas: Ensure the correct formulas for each calculation are used. For example, the difference between population and sample formulas for variance.
- Miscalculating Means: Errors in the calculation of averages can lead to incorrect covariance and correlation results.
