/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Find the equation of the least-s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 22: Problem 4

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. $$\begin{array}{l|r|r|r|r|r|r|r|r|r|r}x & 1 & 3 & 6 & 5 & 8 & 10 & 4 & 7 & 3 & 8 \\\\\hline y & 15 & 12 & 10 & 8 & 9 & 2 & 11 & 9 & 11 & 7 \end{array}$$

Short Answer

Expert verified
The equation of the least-squares line is \( y = 15.0 - 1.018x \).

Step by step solution

01

Calculate means

We'll start by calculating the mean of the x-values and the y-values. The mean of x, denoted as \( \overline{x} \), is computed as:\[\overline{x} = \frac{\sum{x_i}}{n} = \frac{1 + 3 + 6 + 5 + 8 + 10 + 4 + 7 + 3 + 8}{10} = 5.5\]The mean of y, denoted as \( \overline{y} \), is computed as:\[\overline{y} = \frac{\sum{y_i}}{n} = \frac{15 + 12 + 10 + 8 + 9 + 2 + 11 + 9 + 11 + 7}{10} = 9.4\]
02

Calculate slope (b)

The next step is to compute the slope \( b \) of the least-squares line using the formula:\[b = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}}\]Calculate \( \sum{(x_i - \overline{x})(y_i - \overline{y})} \):\[(1-5.5)(15-9.4) + (3-5.5)(12-9.4) + (6-5.5)(10-9.4) + \ldots + (8-5.5)(7-9.4) = -84\]Calculate \( \sum{(x_i - \overline{x})^2} \):\[(1-5.5)^2 + (3-5.5)^2 + (6-5.5)^2 + \ldots + (8-5.5)^2 = 82.5\]Thus,\[b = \frac{-84}{82.5} \approx -1.018\]
03

Calculate y-intercept (a)

Using the slope \( b = -1.018 \), the y-intercept \( a \) can be calculated using the formula:\[a = \overline{y} - b \overline{x}\]Substitute the known values:\[a = 9.4 - (-1.018)(5.5) = 9.4 + 5.599 = 15.0\]
04

Write the equation

The equation of the least-squares line is given by:\[y = a + bx\]Substituting the computed values for \( a \) and \( b \):\[y = 15.0 - 1.018x\]
05

Graph the data and the line

Graph the data points and the least-squares line on the same plot. Plot the x-values (1, 3, 6, 5, 8, 10, 4, 7, 3, 8) against their corresponding y-values (15, 12, 10, 8, 9, 2, 11, 9, 11, 7). Then, draw the line represented by the equation \( y = 15.0 - 1.018x \) across the same range of x-values. Check that the line closely follows the trend of the points but may not pass through any particular points.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Slope Calculation
Calculating the slope is an essential part of determining the least-squares regression line. The slope, denoted as \( b \), measures how much the dependent variable \( y \) changes for a one-unit change in the independent variable \( x \). To compute the slope, use the formula:
  • \[ b = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}} \]
This formula involves two primary calculations:
  • How much each individual \( x \)-value differs from the mean of \( x \), \( \overline{x} \), and the same for \( y \).
  • The product of these differences for each data point, summed up over all data points.
Divide this by the sum of the squares of the differences for \( x \). This result, \( b = -1.018 \), indicates that for each unit increase in \( x \), \( y \) is expected to decrease by about 1.018 units.
Y-Intercept
The y-intercept, \( a \), is the point where the regression line intersects the y-axis. It represents the value of \( y \) when \( x = 0 \). In the least-squares regression context, it is calculated using the formula:
  • \[ a = \overline{y} - b\overline{x} \]
After substituting the calculated mean values and the slope into this formula, \( a \) becomes \( 15.0 \). This means when the independent variable \( x \) is zero, the dependent variable \( y \) is expected to be 15.0. Remember, this calculation uses the average values to ensure the line minimizes the total distance to all the data points.
Mean Calculation
Finding the mean values of \( x \) and \( y \) is a foundational step in calculating the least-squares regression line. The mean, or average, of the data points simplifies them into a single representative value.
  • The arithmetic mean for \( x \) is calculated as \[ \overline{x} = \frac{\sum{x_i}}{n} = 5.5 \]
  • Similarly, the arithmetic mean for \( y \) is \[ \overline{y} = \frac{\sum{y_i}}{n} = 9.4 \]
This involves summing all x or y values and dividing by the number of observations \( n \). The means are crucial to determine both the slope and y-intercept of the regression line, as they represent the central location of the data.
Graphing Data Points
Visualizing the data points and the regression line on a graph is essential for understanding the relationship between variables. Each data point represents the original values of \( x \) and \( y \).
  • Plot each \( (x, y) \) pair on a coordinate plane.
  • Draw the regression line, \( y = 15.0 - 1.018x \), using the calculated slope and y-intercept.
This graph helps to visually assess how well the line fits the data:
  • Notice that while the line may not pass through each point, it trends along the general direction of the data.
  • The degree to which the line captures the spread of points indicates the strength of the linear relationship.
By plotting the line and data together, it鈥檚 easier to see the least-squares method at work, offering a clear illustration of the variable relationships.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the indicated quantities.The alcohol contents (in \(\%\) ) in the blood- streams of those charged with DUI at a police check point were as follows:$$\begin{aligned} &0.15,0.11,0.13,0.16,0.12,0.09,0.10,0.11,0.13,0.06,0.12\\\ &0.11,0.09,0.17,0.14,0.15,0.11,0.14,0.12,0.15,0.10,0.11,0.13 \end{aligned}$$.Draw a frequency polygon using six classes.

Use the following data. The lifetimes of a certain type of automobile tire have been found to be distributed normally with a mean lifetime of \(100,000 \mathrm{km}\) and a standard deviation of \(10,000 \mathrm{km}\) Answer the following questions for a sample of 5000 of these tires. What percent of the samples of 5000 of these tires should have a mean standard deviation of between \(9900 \mathrm{km}\) and \(10,100 \mathrm{km} ?\)

Use the following sets of numbers. A: 3,6,4,2,5,4,7,6,3,4,6,4,5,7,3 B: 25,26,23,24,25,28,26,27,23,28,25 C: 0.48,0.53,0.49,0.45,0.55,0.49,0.47,0.55,0.48,0.57, 0.51,0.46,0.53,0.50,0.49,0.53 D: 105,108,103,108,106,104,109,104,110,108,108, 104,113,106,107,106,107,109,105,111,109,108 Determine the median of the numbers of the given set. Set \(D\)

Find the equation of the indicated least squares curve. Sketch the curve and plot the data points on the same graph. For the points in the following table, find the least-squares curve \(y=m \sqrt{x}+b\). $$\begin{array}{l|c|c|c|c|c} x & 0 & 4 & 8 & 12 & 16 \\\\\hline y & 1 & 9 & 11 & 14 & 15\end{array}$$

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. In an electrical experiment, the following data were found for the values of current and voltage for a particular element of the circuit. Find the voltage \(V\) as a function of the current \(i\). $$\begin{array}{l|r|r|r|r|r}\text {Current }(\mathrm{mA}) & 15.0 & 10.8 & 9.30 & 3.55 & 4.60 \\\\\hline \text {Voltage}(\mathrm{V}) & 3.00 & 4.10 & 5.60 & 8.00 & 10.50\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.