Problem 15 - Algebra 1

Question

Draw a scatter plot of the data. Draw a line that corresponds closely to the data and write an equation of the line. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 3.0 & 9.9 \\ \hline 3.5 & 9.7 \\ \hline 3.7 & 8.6 \\ \hline 4.0 & 8.1 \\ \hline 4.0 & 8.4 \\ \hline 4.5 & 7.4 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

After plotting the points on a scatter plot and drawing an estimated 'best fit' line, the slope of the line and y-intercept are calculated. These values are then incorporated into the linear equation, resulting in the final equation of the line approximating the data points.

Step 1: Plotting a scatter plot

First, draw two axes on a graphing paper or a software, denoting x and y respectively. Then plot the given points (x,y) on the graph to create a scatter plot. The points would be: (3.0,9.9), (3.5,9.7), (3.7,8.6), (4.0,8.1), (4.0,8.4), and (4.5,7.4).

Step 2: Estimation of 'best fit' line

After having a scatter plot, draw a line that appears to fit the data points fairly well. This line doesn't need to touch every data point but must be close. It might be subject to alteration in subsequent steps.

Step 3: Calculation of slope

Calculate the slope of the regression line i.e., \(\Delta y / \Delta x\). The slope will be the increment in y for each unit increment in x. This can be done by finding the difference in Y values divided by the difference in corresponding X values. This method is used commonly in predicting y given x.

Step 4: Calculation of y-intercept

Calculate the y-intercept, which can be found by placing the calculated slope and a known point from the dataset into the equation and solving for b. Y-intercept (b) is the value of y when x = 0

Step 5: Establishing the equation

After having the slope (m) and y-intercept (b), put these values into the Linear equation, which is y = mx + b. This will be the final equation of the line.

Key Concepts

Creating Scatter Plots

A scatter plot is a type of graph that represents individual data points on a two-dimensional plane. It's a powerful tool for visualizing the relationship between two variables.

To create a scatter plot, you start by drawing two perpendicular lines, which are the axes. The horizontal axis usually represents the independent variable, while the vertical axis represents the dependent variable. After determining the scale based on the data range, you plot the points by marking a dot where the x and y values meet on the graph.

For instance, if you have a data point (3.0, 9.9), you find 3.0 on the x-axis and 9.9 on the y-axis, then plot a point where these two values intersect. Repeat this for all data points to complete the scatter plot. It's important that the points are plotted accurately, as they form the basis for further analysis.

Finding Slope of a Line

The slope of a line is a measure of how steep it is and is calculated as the ratio of the change in the y-value to the change in the x-value, known as 'rise over run'. Mathematically, it's expressed as \( m = \frac{\Delta y}{\Delta x} \).

To find the slope between two points, you take the difference in y-values and divide it by the difference in x-values. For example, if you have two points (x1, y1) and (x2, y2), the slope m is computed as \( m = \frac{y2 - y1}{x2 - x1} \).

Determining an accurate slope is crucial when creating a line of best fit on a scatter plot because it affects how well the line describes the data.

Linear Regression Analysis

Linear regression analysis is a method used to find the best-fitting line through a set of data points in a scatter plot. This line of best fit allows us to predict values and ascertain the strength of the relationship between the two variables.

It involves finding the equation of the line, usually of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The regression line minimizes the distance between itself and all the points on the scatter plot, providing a visual representation of the data's trend.

The process usually utilizes calculation methods such as the least squares method to minimize these distances, which are squared and summed to find the line that has the smallest possible sum, the line of best fit.

Y-Intercept Calculation

The y-intercept of a line is the point where the line crosses the y-axis, which is when the value of x is zero. It's denoted as \( b \) in the linear equation \( y = mx + b \). To calculate the y-intercept, we need the slope \( m \) and at least one known point on the line.

Take a known point \( (x_1, y_1) \) and the slope, and plug them into the linear equation to solve for \( b \): \( y_1 = mx_1 + b \). Rearrange the terms to \( b = y_1 - mx_1 \), substituting the point's coordinates and the computed slope. This value is key to anchoring the line at the correct position on the graph.

Writing Linear Equations

Once you've determined the slope \( m \) and y-intercept \( b \), you can write the equation of the line. A linear equation allows us to model the relationship between two variables with a line and is expressed in the form \( y = mx + b \).

If, for example, we found a slope of -0.5 and a y-intercept of 10, the linear equation would be written as \( y = -0.5x + 10 \). We can use this equation to predict the value of y for any given x within the data range. Writing the correct linear equation is fundamental for making accurate predictions and conclusions from our data.