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.