Subjects statistics

Least Squares Regression

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Search Solutions

Least Squares Regression


1. **Problem Statement:** We need to find the least-squares regression line for predicting the distance a baseball travels (response variable $y$) based on the speed at which it was hit (explanatory variable $x$). 2. **Formula:** The least-squares regression line is given by: $$\hat{y} = b x + a$$ where $b$ is the slope and $a$ is the y-intercept. 3. **Calculating the slope $b$:** The slope is calculated by: $$b = r \times \frac{s_y}{s_x}$$ where $r$ is the correlation coefficient between $x$ and $y$, $s_y$ is the standard deviation of $y$, and $s_x$ is the standard deviation of $x$. 4. **Calculating the intercept $a$:** The intercept is calculated by: $$a = \bar{y} - b \bar{x}$$ where $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$ respectively. 5. **Interpretation:** The regression line predicts the distance the ball travels based on the speed it was hit. 6. **Note:** Since the actual data values, means, standard deviations, and correlation coefficient are not provided here, you need to calculate or obtain these values from the data to compute $b$ and $a$. 7. **Final answer format:** $$\hat{y} = b x + a$$ with $b$ and $a$ rounded to three decimal places once calculated.