To analyze the relationship between the percentage of three-point shots attempted (x) and the number of wins (y), we can use linear regression to find the equation of the line that best fits the data. Given the data points, we can find the slope (b) and the y-intercept (a) for the line in the form:
\[ y = bx + a \]
Data Points
- (22, 22)
- (26, 36)
- (28, 40)
- (29, 46)
- (30, 47)
Step 1: Calculate the Means
Calculate the mean of x and y:
\[ \bar{x} = \frac{22 + 26 + 28 + 29 + 30}{5} = \frac{135}{5} = 27 \] \[ \bar{y} = \frac{22 + 36 + 40 + 46 + 47}{5} = \frac{191}{5} = 38.2 \]
Step 2: Calculate the Slope (b)
Using the formula for the slope \( b \):
\[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
Calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \)
| \( x \) | \( y \) | \( x - \bar{x} \) | \( y - \bar{y} \) | \( (x - \bar{x})(y - \bar{y}) \) | |---------|---------|--------------------|--------------------|---------------------------------| | 22 | 22 | -5 | -16.2 | 81.0 | | 26 | 36 | -1 | -2.2 | 2.2 | | 28 | 40 | 1 | 1.8 | 1.8 | | 29 | 46 | 2 | 7.8 | 15.6 | | 30 | 47 | 3 | 8.8 | 26.4 |
\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 81.0 + 2.2 + 1.8 + 15.6 + 26.4 = 126.0 \]
Calculate \( \sum (x_i - \bar{x})^2 \)
| \( x \) | \( x - \bar{x} \) | \( (x - \bar{x})^2 \) | |---------|--------------------|------------------------| | 22 | -5 | 25 | | 26 | -1 | 1 | | 28 | 1 | 1 | | 29 | 2 | 4 | | 30 | 3 | 9 |
\[ \sum (x_i - \bar{x})^2 = 25 + 1 + 1 + 4 + 9 = 40 \]
Step 3: Calculate \( b \)
\[ b = \frac{126.0}{40} = 3.15 \]
Step 4: Calculate the Intercept (a)
Using the formula for the y-intercept \( a \):
\[ a = \bar{y} - b\bar{x} \] \[ a = 38.2 - (3.15 \times 27) = 38.2 - 85.05 = -46.85 \]
Final Equation
Thus, the equation of the line is:
\[ y = 3.15x - 46.85 \]
Answer
Rounding to the nearest thousandth:
\[ y = 3.150x - 46.850 \]