Math Problem Statement

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?

Solution

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))
12
23
35
44
57

The question asks for the linear function that best models the data. The options provided are:

  1. f(x)=1.9x+1.1f(x) = 1.9x + 1.1
  2. f(x)=0.9x+1.1f(x) = 0.9x + 1.1
  3. f(x)=1.1x+0.9f(x) = 1.1x + 0.9
  4. f(x)=1.1x+1.9f(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+bf(x) = mx + b.

Let’s calculate the slope first:

The slope mm can be calculated by using the formula:

m=ni=1n(xiyi)(i=1nxi)(i=1nyi)ni=1nxi2(i=1nxi)2m = \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:

  • xix_i represents the day,
  • yiy_i represents the number of miles,
  • nn 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+0.9f(x) = 1.1x + 0.9

This matches the third option in the question:

f(x)=1.1x+0.9f(x) = 1.1x + 0.9

Therefore, the best model for the data is:

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

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Algebra
Slope-Intercept Form

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Linear equation: f(x) = mx + b

Theorems

Linear Regression Best-Fit Line

Suitable Grade Level

Grades 8-10