https://file.aiquickdraw.com/mathbot/file/1734141363676xra6f1iy.jpg

To solve this problem, we will compute the least-squares regression line \( y = mx + b \) using the data provided in the table. The slope \( m \) and \( y \)-intercept \( b \) are calculated using the following formulas:

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

\[
b = \bar{y} - m\bar{x}
\]

Where:
- \( x_i \) is the father's height,
- \( y_i \) is the son's height,
- \( \bar{x} \) is the mean of father's heights,
- \( \bar{y} \) is the mean of son's heights.

Let’s calculate step by step. numpy as np

# Data from the table
fathers_height = np.array([65.4, 65.7, 66.7, 70.1, 72.4, 65.7, 72.3, 73.6])
sons_height = np.array([66, 70.9, 68.8, 73.3, 69.1, 65.3, 71.9, 76.5])

# Calculate means
x_mean = np.mean(fathers_height)
y_mean = np.mean(sons_height)

# Calculate the slope (m)
numerator = np.sum((fathers_height - x_mean) * (sons_height - y_mean))
denominator = np.sum((fathers_height - x_mean)**2)
slope = numerator / denominator

# Calculate the y-intercept (b)
intercept = y_mean - slope * x_mean

slope, interceptThe least-squares regression line for predicting the son's height (\( y \)) from the father's height (\( x \)) is:

\[
y = 0.7833x + 16.1867
\]

Would you like a detailed explanation of the calculation process? 

Here are some related questions to explore:
1. How do you compute the correlation coefficient \( r \) for the given data?
2. What does the slope of the regression line signify in this context?
3. How can we use the regression line to predict a son's height given a specific father's height?
4. What is the coefficient of determination (\( r^2 \)), and how is it interpreted here?
5. How sensitive is the regression line to outliers in the data?

**Tip:** Always plot the data along with the regression line to visualize the fit and detect any anomalies.

Compute the least-squares regression line for predicting a son's height (y) from a father's height (x) using the provided table data. Round the slope and y-intercept values to at least four decimal places.

Learn how to compute the least-squares regression line for father-son height data. This step-by-step solution uses statistical concepts and key formulas to find the slope (0.7833) and y-intercept (16.1867). Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation