Questions: Regression Equation Homework Grade (%) Test Grade (%) 59 50 69 64 80 75 75 64 51 46 88 69 78 70 Correlation coefficient r:

Solution Steps

To find the correlation coefficient \( r \) between two sets of data, we can use the Pearson correlation formula. This involves calculating the covariance of the two variables and dividing it by the product of their standard deviations. The correlation coefficient will indicate the strength and direction of the linear relationship between the two variables.

We have two sets of data: homework grades and test grades. The homework grades are \([59, 69, 80, 75, 51, 88, 78]\) and the test grades are \([50, 64, 75, 64, 46, 69, 70]\).

The correlation coefficient \( r \) is calculated using the Pearson correlation formula, which is given by:

\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]

where \( x_i \) and \( y_i \) are the individual data points, and \( \bar{x} \) and \( \bar{y} \) are the means of the homework and test grades, respectively.

The calculated correlation coefficient is \( r = 0.9229 \). This value is close to 1, indicating a strong positive linear relationship between homework grades and test grades. This suggests that as homework grades increase, test grades tend to increase as well.

Final Answer