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It looks like you have uploaded a multiple-choice question regarding a dataset of miles jogged each day. Based on the given data, I will assist in identifying the linear best-fit function.

Here’s the data provided:

| Day (x) | Number of Miles (f(x)) |
|---------|------------------------|
| 1       | 2                      |
| 2       | 3                      |
| 3       | 5                      |
| 4       | 4                      |
| 5       | 7                      |

The question asks for the linear function that best models the data. The options provided are:
1. \( f(x) = 1.9x + 1.1 \)
2. \( f(x) = 0.9x + 1.1 \)
3. \( f(x) = 1.1x + 0.9 \)
4. \( f(x) = 1.1x + 1.9 \)

To find the best model, we’ll calculate the slope (m) and y-intercept (b) for a linear equation of the form \( f(x) = mx + b \).

Let’s calculate the slope first:

The slope \( m \) can be calculated by using the formula:

\[
m = \frac{n \sum_{i=1}^{n} (x_i y_i) - (\sum_{i=1}^{n} x_i)(\sum_{i=1}^{n} y_i)}{n \sum_{i=1}^{n} x_i^2 - (\sum_{i=1}^{n} x_i)^2}
\]

Where:
- \( x_i \) represents the day,
- \( y_i \) represents the number of miles,
- \( n \) is the total number of data points.

Let me calculate the best-fit line for this data. numpy as np

# Given data
x = np.array([1, 2, 3, 4, 5])  # Days
y = np.array([2, 3, 5, 4, 7])  # Miles jogged

# Calculate the best-fit line (linear regression)
A = np.vstack([x, np.ones(len(x))]).T
m, b = np.linalg.lstsq(A, y, rcond=None)[0]

m, bThe linear regression for the given data provides the equation:

\[
f(x) = 1.1x + 0.9
\]

This matches the third option in the question:

\[
f(x) = 1.1x + 0.9
\]

Therefore, the best model for the data is:

**Option (3): \( f(x) = 1.1x + 0.9 \)**.

Would you like further details on how the regression was done, or do you have any additional questions? 

Here are 5 related questions you might find useful:
1. How can I calculate the error or residuals in a linear regression?
2. What are the steps involved in performing linear regression manually?
3. How do you interpret the slope and intercept of a linear equation?
4. How can we assess the quality of a linear regression fit?
5. What is the difference between linear and nonlinear regression models?

Tip: When working with regression, always plot the data and the regression line to visually assess the fit before concluding.

Jorge recorded the number of miles, f(x), he jogs each day over the last 5 days. The data is modeled with the linear best-fit function. Which function best models the data set?

Determine the best-fit linear function for Jorge's jogging data. Analyze the dataset of miles jogged over 5 days and identify the correct equation using linear regression techniques. Suitable for Grades 8-10 students studying linear functions.

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