https://file.aiquickdraw.com/mathbot/file/1727072385772eb1vfgm4.jpg

The problem involves calculating the equation of a regression line based on the given data and then using this equation to predict the murder rates for specific states with different numbers of registered automatic weapons.

We are given the following dataThe problem involves calculating the equation of a regression line based on the given data and then using this equation to predict the murder rates for specific states with different numbers of registered automatic weapons.

We are given the following 
- \( x \) = thousands of automatic weapons
- \( y \) = murders per 100,000 residents

| \( x \) | 11.5 | 8.4 | 7.1 | 3.3 | 2.9 | 2.4 | 2.6 | 0.6 |
|--------|------|-----|-----|-----|-----|-----|-----|-----|
| \( y \) | 13.9 | 11.4 | 10.3 | 6.6 | 6.8 | 6.1 | 6.1 | 4.7 |

### Steps:
1. **Find the regression line equation \( y = ax + b \)**: 
   - We'll calculate the slope \( a \) and intercept \( b \) using the least squares method.

2. **Use the regression line** to:
   - Predict the number of murders per 100,000 residents for a state with:
     - 9.9 thousand automatic weapons
     - 4.5 thousand automatic weapons

Let me calculate the regression equation for this data first.The regression equation is:

\[
y = 0.86x + 4.05
\]

Where:
- \( y \) is the predicted number of murders per 100,000 residents.
- \( x \) is the number of automatic weapons in thousands.

Now, let's use this equation to answer the two questions:

### A) Predict the number of murders per 100,000 residents for a state with 9.9 thousand automatic weapons.

Substitute \( x = 9.9 \) into the equation.

### B) Predict the number of murders per 100,000 residents for a state with 4.5 thousand automatic weapons.

Substitute \( x = 4.5 \) into the equation.

I'll calculate both values next.### A) The predicted number of murders per 100,000 residents for a state with 9.9 thousand automatic weapons is **12.598** (rounded to 3 decimal places).

### B) The predicted number of murders per 100,000 residents for a state with 4.5 thousand automatic weapons is **7.935** (rounded to 3 decimal places).

Would you like more details on how these values were derived, or do you have any questions?

Here are 5 related questions to explore further:
1. How is the slope of the regression line interpreted in the context of this data?
2. What does the correlation coefficient (\(r\)) tell us about the relationship between weapons and murder rates?
3. How would the prediction change if you had a different data set or more data points?
4. What would the prediction be for a state with 0 registered automatic weapons?
5. How does interpolation differ from extrapolation in this regression context?

**Tip:** Always check the residuals of your regression model to ensure it fits well with the given data.

The table shows the number of state-registered automatic weapons and the murder rate for several Northwestern states. Use this data to find the equation of the regression line in y = ax + b form. Then predict the number of murders per 100,000 residents for a state with 9.9 thousand automatic weapons and 4.5 thousand automatic weapons.

Learn how to calculate the regression equation for predicting murder rates based on the number of state-registered automatic weapons. Using statistical data, this problem demonstrates the application of linear regression to make predictions about the murder rate for states with varying numbers of automatic weapons. Suitable for advanced high school and college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation