/*! 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 27 Explain why the slope \(b\) of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 27

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r .\)

Short Answer

Expert verified
The slope of the least-squares line and the sample correlation coefficient always have the same sign because both are indicative of the direction of the relationship between the two variables in a linear model. A positive sign indicates that one variable increases as the other also increases, and a negative sign signifies that as one variable increases, the other decreases.

Step by step solution

01

Understanding the Basics

Recall that the sample correlation coefficient, \(r\), measures the strength and direction of a linear relationship between two variables. It is a value between -1 and 1. A negative \(r\) indicates a negative linear relationship, a positive \(r\) indicates a positive linear relationship, and \(r=0\) implies no linear relationship.
02

Defining the Slope

The slope \(b\) of the least-squares line also implies the direction of the relationship between the two variables. A positive slope suggests that as one variable increases, the other also increases, and similarly, a negative slope implies that as one variable increases, the other decreases.
03

Draw the Connection

Because both \(r\) and \(b\) are indicative of the direction of the relationship between the two variables in a linear model, their signs will always match. If \(r\) is positive, \(b\) will be positive, indicating that one increases as the other increases in the model. If \(r\) is negative, \(b\) will be negative, signifying that one value decreases as the other increases in the model.

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.

Least-Squares Line
The least-squares line, often called the line of best fit, is a straight line that best represents the data on a scatter plot. It is calculated through a method that minimizes the sum of the squared differences, or **squared residuals**, between the observed values and the values predicted by the line. This ensures that the line is as close to all data points as possible, thereby reducing error.

Key characteristics of a least-squares line include:
  • It showcases the trend in the data by depicting the average direction of the relationship between the independent and dependent variables.
  • The position and slope of this line provide insights into patterns and trends, helping in predictions based on historical data.
  • This line is used in various fields such as economics, biology, and engineering for modeling and predicting outcomes.
Correlation Coefficient
The correlation coefficient, denoted as **r**, is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where:

  • **r = 1** indicates a perfect positive linear correlation.
  • **r = -1** indicates a perfect negative linear correlation.
  • **r = 0** suggests no linear relationship between the variables.
This coefficient helps to quantify the degree to which the variables move in relation to each other.

Several important points about the correlation coefficient include:
  • A positive **r** value implies that as one variable increases, the other variable tends to increase as well.
  • A negative **r** value suggests an inverse relationship, meaning that as one variable increases, the other decreases.
  • This measure is dimensionless, making it easy to compare across different datasets and contexts.
Slope of a Line
The slope of a line is a numerical value that describes the steepness, direction, and rate of change between two variables on a graph. It is often represented by the letter **b** in equations of the line, like the least-squares line.

Here's how the slope functions:
  • A **positive slope** indicates that as the x-variable (independent variable) increases, the y-variable (dependent variable) also increases. This signifies a positive relationship.
  • A **negative slope** means that as the x-variable increases, the y-variable decreases, reflecting a negative relationship between the variables.
  • A **slope of zero** implies no change, meaning the line is horizontal, showing that the dependent variable does not change with the independent variable.
Understanding the slope is crucial because it provides insight into the magnitude and nature of the change being modeled in a linear regression.

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

The sales manager of a large company selected a random sample of \(n=10\) salespeople and determined for each one the values of \(x=\) years of sales experience and \(y=\) annual sales (in thousands of dollars). A scatterplot of the resulting \((x, y)\) pairs showed a linear pattern. a. Suppose that the sample correlation coefficient is \(r=.75\) and that the average annual sales is \(\bar{y}=100 .\) If a particular salesperson is 2 standard deviations above the mean in terms of experience, what would you predict for that person's annual sales? b. If a particular person whose sales experience is 1\. 5 standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of \(r\) ?

Determining the age of an animal can sometimes be a difficult task. One method of estimating the age of harp seals is based on the width of the pulp canal in the seal's canine teeth. To investigate the relationship between age and the width of the pulp canal, researchers measured age and canal width in seals of known age. The following data on \(x=\) age (in years) and \(y=\) canal length (in millimeters) are a portion of a larger data set that appeared in the paper "Validation of Age Estimation in the Harp Seal Using Dentinal Annuli" (Canadian Journal of Fisheries and Aquatic Science [1983]: \(1430-1441):\) \(\begin{array}{rrrrrrrrr}x & 0.25 & 0.25 & 0.50 & 0.50 & 0.50 & 0.75 & 0.75 & 1.00 \\ y & 700 & 675 & 525 & 500 & 400 & 350 & 300 & 300 \\ x & 1.00 & 1.00 & 1.00 & 1.00 & 1.25 & 1.25 & 1.50 & 1.50 \\ y & 250 & 230 & 150 & 100 & 200 & 100 & 100 & 125 \\ x & 2.00 & 2.00 & 2.50 & 2.75 & 3.00 & 4.00 & 4.00 & 5.00 \\ y & 60 & 140 & 60 & 50 & 10 & 10 & 10 & 10 \\ x & 5.00 & 5.00 & 5.00 & 6.00 & 6.00 & & & \\ y & 15 & 10 & 10 & 15 & 10 & & & \end{array}\) Construct a scatterplot for this data set. Would you describe the relationship between age and canal length as linear? If not, suggest a transformation that might straighten the plot.

The article "Examined Life: What Stanley H. Kaplan Taught Us About the SAT" (The New yorker [December 17. 2001\(]: 86-92\) ) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict firstyear college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at 15.4 percent, and SAT I was last at 13.3 percent." a. If the data from this study were used to fit a leastsquares line with \(y=\) first-year college GPA and \(x=\) high school GPA, what would be the value of \(r^{2} ?\) b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would be very accurate? Explain why or why not.

Some plant viruses are spread by insects and tend to spread from the edges of a field inward. The data on \(x=\) distance from the edge of the field (in meters) and \(y=\) proportion of plants with virus symptoms that appeared in the paper "Patterns of Spread of Two NonPersistently Aphid-Borne Viruses in Lupin Stands" \((A n-\) nals of Applied Biology [2005]: \(337-350\) ) was used to fit a least-squares regression line to describe the relationship between \(x\) and \(y^{\prime}=\ln \left(\frac{p}{1-p}\right) .\) Minitab output resulting from fitting the least-squares line is given below. The regression equation is \(\ln (p /(1-p))=-0.917-0.107\) Distance to Crop Edge \(\begin{array}{lrrrr}\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P. } \\ \text { Constant } & -0.9171 & 0.1249 & -7.34 & 0.000\end{array}\) \begin{tabular}{lrrrr} Constant & -0.9171 & 0.1249 & -7.34 & 0.000 \\ Distance to Crop Edge & -0.10716 & 0.01062 & -10.09 & 0.000 \\ \hline \end{tabular} \(\mathrm{S}=0.387646 \quad \mathrm{R}-\mathrm{Sq}=72.8 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=72.1 \%\) a. What is the logistic regression function relating \(x\) and the proportion of plants with virus symptoms? b. What would you predict for the proportion of plants with virus symptoms at a distance of 15 meters from the edge of the field? (Note: the \(x\) values in the data set ranged from 0 to \(20 .)\)

Draw two scatterplots, one for which \(r=1\) and a second for which \(r=-1\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.