Finding The Line Of Best Fit For Raquel's Dart Throws A Mathematical Solution

Let's dive into a fun mathematical problem involving Raquel, her darts, and a coordinate grid! Raquel's throwing darts at a coordinate grid centered at the origin, and she's aiming to create a line with her darts. However, darts don't always land perfectly, so we end up with a scatter of points. Our mission is to find the equation that best represents the line Raquel is trying to create. This is a classic problem in statistics and data analysis, where we use the concept of the line of best fit to model a relationship between variables. So, let's put on our mathematical hats and figure out which equation best approximates the line of best fit for Raquel's dart throws!

Understanding the Problem

Before we jump into calculations, let's make sure we understand the situation. Raquel has thrown six darts, and they've landed at these coordinates:

• (-5, 0)
• (1, -3)
• (4, 5)
• (-8, -6)
• (0, 2)
• (9, 6)

These points don't form a perfect line, but we can see a general trend. Our goal is to find a line that comes as close as possible to all of these points. This line is called the line of best fit, also known as the least squares regression line. It's the line that minimizes the sum of the squares of the vertical distances between the points and the line. This might sound a bit complicated, but don't worry, we'll break it down step by step.

Finding the Line of Best Fit: A Step-by-Step Guide

There are a few ways to find the line of best fit. We will focus on the manual calculation method, which involves some statistics but gives a solid understanding of the process. The equation of a line is generally represented as y = mx + b, where m is the slope and b is the y-intercept. So, our goal is to find the values of m and b that give us the best fit for our data points.

Step 1: Calculate the Means (Averages)

The first step is to calculate the mean (average) of the x-coordinates and the mean of the y-coordinates. This will give us a central point that the line of best fit will pass through.

Mean of x-coordinates (x̄):

Add up all the x-coordinates and divide by the number of points (which is 6 in our case):

x̄ = (-5 + 1 + 4 + (-8) + 0 + 9) / 6 = 1 / 6 ≈ 0.167

Mean of y-coordinates (ȳ):

Do the same for the y-coordinates:

ȳ = (0 + (-3) + 5 + (-6) + 2 + 6) / 6 = 4 / 6 ≈ 0.667

So, the mean point is approximately (0.167, 0.667). This point is crucial because the line of best fit will always pass through it.

Step 2: Calculate the Slope (m)

The slope (m) tells us how steep the line is. The formula for the slope of the line of best fit is:

m = Σ[(xi - x̄)(yi - ȳ)] / Σ[(xi - x̄)²]

Where:

• xi is the x-coordinate of each point
• yi is the y-coordinate of each point
• x̄ is the mean of the x-coordinates
• ȳ is the mean of the y-coordinates
• Σ means "sum of"

This formula might look intimidating, but let's break it down. We need to calculate two sums:

1. The sum of the products of (xi - x̄) and (yi - ȳ) for each point.
2. The sum of the squares of (xi - x̄) for each point.

Let's create a table to help us with these calculations:

Now we can plug these sums into the slope formula:

m = 118.421 / 186.833 ≈ 0.634

So, the slope of our line of best fit is approximately 0.634.

Step 3: Calculate the Y-intercept (b)

The y-intercept (b) is the point where the line crosses the y-axis. We can find it using the following formula:

b = ȳ - m * x̄

We already have the values for ȳ, m, and x̄:

b = 0.667 - 0.634 * 0.167 ≈ 0.667 - 0.106 ≈ 0.561

So, the y-intercept is approximately 0.561.

Step 4: Write the Equation of the Line

Now that we have the slope (m) and the y-intercept (b), we can write the equation of the line of best fit:

y = mx + b

y ≈ 0.634x + 0.561

Conclusion: The Best Fit for Raquel's Darts

After our calculations, we've found that the equation that best approximates the line of best fit for Raquel's dart throws is approximately y = 0.634x + 0.561. This means that, on average, for every one unit increase in the x-coordinate, the y-coordinate increases by about 0.634, and the line crosses the y-axis at around 0.561.

This exercise demonstrates how we can use mathematical tools to analyze real-world data. Even though Raquel's darts didn't land perfectly on a line, we were able to find a line that best represents the overall trend of her throws. This technique is used in many fields, from predicting stock prices to understanding scientific data. So, the next time you see a scatter of points, remember that the line of best fit can help you make sense of it all!

Keywords: line of best fit, least squares regression, slope, y-intercept, coordinate grid, data analysis, mathematics, statistics, mean, average