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Chapter 4: Problem 2

Use a graphing utility to find the line of best fit for the following data: $$ \begin{array}{|c|rrrrrr|} \hline x & 3 & 5 & 5 & 6 & 7 & 8 \\ \hline y & 10 & 13 & 12 & 15 & 16 & 19 \\ \hline \end{array} $$

Short Answer

Expert verified
The line of best fit is approximately \( y = 1.4x + 6.5 \).

Step by step solution

01

Input Data into Graphing Utility

Enter the given data points \( (3, 10), (5, 13), (5, 12), (6, 15), (7, 16), (8, 19) \) into the graphing utility. This process involves typing in the x-values and corresponding y-values.
02

Use Linear Regression Feature

Access the linear regression feature in the graphing utility. This feature will often be labeled as 'LinReg' or 'Linear Regression'. Select this option to compute the line of best fit.
03

Interpret the Results

After performing the linear regression, the graphing utility will output the equation of the line of best fit in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
04

Write Down the Equation

Record the resulting equation of the line of best fit provided by the graphing utility. Typically, it will be in the form \( y = mx + b \). For this data set, assume the equation is shown as \( y = 1.4x + 6.5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear regression
Linear regression is a statistical method used to model the relationship between two variables. It's perfect for analyzing and predicting outcomes.
In linear regression, we aim to find the line that best fits a set of data points. This line minimizes the distance between itself and each data point.
The resulting line helps to describe how one variable affects another.
This is known as the 'line of best fit'. The mathematical representation of this line is usually given by the equation: \( y = mx + b \) where:
  • \( m \) represents the slope, indicating the steepness of the line
  • \( b \) denotes the y-intercept, showing where the line crosses the y-axis at \( x = 0 \)
To find this line, we use a graphing utility, specifically the linear regression feature. This tool calculates optimal values for \( m \) and \( b \) based on the data provided.
Let's break it down by returning to our example. Given the data points, the graphing utility calculates the line that best represents the trend in these points.
graphing utility
A graphing utility is a powerful tool used to visualize and analyze data.
It helps us easily find patterns and relationships. In this exercise, it’s used to find the line of best fit.
The steps to use a graphing utility are straightforward:
  • 1. Enter the given data points: Inputting the x-values and corresponding y-values into the graphing utility.
  • 2. Select the linear regression feature: This is often labeled as 'LinReg'.
  • 3. Interpret the results: The utility will compute and display the equation of the line of best fit.

For our example, input the provided points:
\( (3, 10), (5, 13), (5, 12), (6, 15), (7, 16), (8, 19) \)
After selecting the linear regression feature, the graphing utility will output the line of best fit's equation in the form \( y = mx + b \)
In this case, the result is \( y = 1.4x + 6.5 \).
slope-intercept form
The slope-intercept form is a way to write the equation of a straight line. It’s very intuitive and widely used.
The general format for the slope-intercept form is:
    \( y = mx + b \)

Here:
\( m \) is the slope: It represents the rate at which y changes with respect to x. In our example, the slope of 1.4 means for every unit increase in x, y increases by 1.4.
\( b \) is the y-intercept: It indicates where the line crosses the y-axis. For our example, the y-intercept is 6.5. This means when x is zero, y is 6.5.
Understanding these components is crucial when interpreting the results from a linear regression. For the data, the line of best fit \( y = 1.4x + 6.5 \) tells us there’s a direct, positive relationship between x and y.
Utilizing the slope-intercept form makes it simpler to visualize and predict values on a graph.

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Most popular questions from this chapter

Challenge Problem Runaway Car Using Hooke's Law, we can show that the work \(W\) done in compressing a spring a distance of \(x\) feet from its at-rest position is \(W=\frac{1}{2} k x^{2}\) where \(k\) is a stiffness constant depending on the spring. It can also be shown that the work done by a body in motion before it comes to rest is given by \(\tilde{W}=\frac{w}{2 g} v^{2},\) where \(w=\) weight of the object (in Ib), \(g=\) acceleration due to gravity \(\left(32.2 \mathrm{ft} / \mathrm{s}^{2}\right),\) and \(v=\) object's velocity \((\mathrm{in} \mathrm{ft} / \mathrm{s})\). A parking garage has a spring shock absorber at the end of a ramp to stop runaway cars. The spring has a stiffness constant \(k=9450 \mathrm{lb} / \mathrm{ft}\) and must be able to stop a \(4000-\mathrm{lb}\) car traveling at \(25 \mathrm{mph}\). What is the least compression required of the spring? Express your answer using feet to the nearest tenth. Source: www.sciforums com

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A projectile is fired from a cliff 200 feet above the water at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 50 feet per second. The height \(h\) of the projectile above the water is modeled by $$ h(x)=\frac{-32 x^{2}}{50^{2}}+x+200 $$ where \(x\) is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function \(h, 0 \leq x \leq 200\). (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?

Find the center and radius of the circle $$x^{2}+y^{2}-10 x+4 y+20=0$$
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