To find the least-squares regression line, we need to calculate the slope ($m$) and the intercept ($b$) of the line, using the formula:

$y = mx + b$

Here’s the process:

1. Calculate the means of the explanatory ($x$) and response ($y$) variables:

   • $\bar{x}$: Mean of the speed (mph)
   • $\bar{y}$: Mean of the distance (feet)

2. Calculate the slope ($m$): $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

3. Calculate the intercept ($b$): $b = \bar{y} - m\bar{x}$

I will calculate these values.import numpy as np import pandas as pd

Data

speed = np.array([103.5, 107.9, 103.4, 103.5, 102.1, 103.0, 103.6, 100.7, 101.3, 101.4, 110.4, 105.3]) distance = np.array([395, 441, 394, 422, 392, 393, 402, 396, 393, 399, 427, 420])

Calculate means

x_mean = np.mean(speed) y_mean = np.mean(distance)

Calculate the slope (m)

numerator = np.sum((speed - x_mean) * (distance - y_mean)) denominator = np.sum((speed - x_mean)**2) slope = numerator / denominator

Calculate the intercept (b)

intercept = y_mean - slope * x_mean

slope, intercept(4.784076403006948, -90.61980048224649)(4.784076403006948, -90.61980048224649)successThe least-squares regression line is:

$y = 4.784x - 90.620$

where:

• $x$ is the speed (mph)
• $y$ is the distance (feet)

Would you like me to plot this line or explain the calculations further?

5 Related Questions:

1. How do we interpret the slope and intercept of this regression line in context?
2. Can you calculate the predicted distance for a ball hit at 105 mph?
3. How well does this regression line fit the data? Would you like to see an $R^2$ value calculation?
4. What are the assumptions underlying the least-squares regression?
5. How do outliers affect the regression line, and are there any in this dataset?

Tip:

Always visualize your data along with the regression line to better understand the fit and identify potential outliers or patterns.