Commute Time (min),x	Well-Being Index score,y
5	68.9
20	67.2
25	66.4
40	65.1
60	63.3
84	61.8
105	59.3 Predict the​ well-being index of a person whose commute time is 30 minutes.
The predicted index score is

enter your response here.
​(Round to one decimal place as​ needed.)

To predict the Well-Being Index score of a person whose commute time is 30 minutes, we can apply linear regression. This process involves finding the equation of the line \( y = mx + b \), where:

- \( x \) represents the commute time,
- \( y \) represents the Well-Being Index score,
- \( m \) is the slope of the line, and
- \( b \) is the y-intercept.

### Step 1: Calculate the slope (m)
The slope of the line is given by the formula:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Where:
- \( n \) is the number of data points,
- \( \sum xy \) is the sum of the products of each \( x \) and \( y \),
- \( \sum x \) is the sum of the commute times,
- \( \sum y \) is the sum of the Well-Being Index scores,
- \( \sum x^2 \) is the sum of the squares of the commute times.

### Step 2: Calculate the y-intercept (b)
The y-intercept is calculated by the formula:
\[
b = \frac{\sum y - m(\sum x)}{n}
\]

### Step 3: Use the equation to predict \( y \) for \( x = 30 \).

Let's calculate this.The predicted Well-Being Index score for a person whose commute time is 30 minutes is approximately **66.3**.

Would you like more details on the process or any clarifications?

Here are 5 related questions to expand on this topic:
1. How is linear regression used to make predictions?
2. What other methods can be used to predict values based on data?
3. How does the slope affect the relationship between two variables?
4. What is the significance of the y-intercept in regression?
5. How do outliers affect linear regression models?

**Tip:** Always check the accuracy of your linear regression model by looking at its residuals or using metrics like R-squared to ensure it fits the data well.

Predict the well-being index of a person whose commute time is 30 minutes.

Learn how to predict the Well-Being Index score of a person whose commute time is 30 minutes using linear regression. This problem involves algebraic concepts like the slope-intercept form and the calculation of the slope and y-intercept. Suitable for students in Grades 9-12.

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