/*! 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 11 Two variables \(x\) and \(y\) ha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 11

Two variables \(x\) and \(y\) have a positive linear relationship. Explain what happens to the value of \(y\) when \(x\) increases. Give one example of a positive relationship between two variables.

Short Answer

Expert verified
In a positive linear relationship, if \(x\) increases, \(y\) will also increase as they move in same direction. An example of such a relationship is the correlation between hours of study and the corresponding test score.

Step by step solution

01

Definition of a Positive Linear Relationship

In a positive linear relationship, two variables, in this case \(x\) and \(y\), change concurrently. It means that when the value of \(x\) goes up, the value of \(y\) also increases.
02

Changes in Variables

Given that there's a positive linear relationship, if we increase the value of \(x\), the value of \(y\) will correspondingly rise. This is a reflection of the positive association between the two variables. If \(x\) goes down, \(y\) will also decrease. The values of both variables move in the same direction.
03

Example of a Positive Relationship

An illustrative example of a positive relationship could be the connection between hours studied and test score. Generally, as the number of hours studied (\(x\)) increases, the test score (\(y\)) tends also to rise.

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.

Correlation
Correlation is a statistical measure that describes the degree and direction of a relationship between two variables. When dealing with a correlation between variables like \(x\) and \(y\), it's essential to understand whether it is positive, negative, or zero.
Positive correlation, which is our focus, means that an increase in one variable corresponds with an increase in the other. For instance, as \(x\) increases, \(y\) also increases.
  • Correlation coefficients range between -1 and 1.
  • A correlation of 1 indicates a perfect positive correlation.
  • A correlation of -1 implies a perfect negative correlation.
  • A correlation of 0 means no correlation at all.
An example of a positive correlation is the relationship between height and weight in humans. Typically, taller people tend to weigh more. Understanding correlation helps in predicting the behavior of variables relative to each other.
Linear Regression
Linear regression is a method used to model the relationship between a dependent variable and one or more independent variables. It aims to establish a linear equation that best fits the data points. The basic form of a linear regression equation is \(y = mx + b\), where:
  • \(y\) is the dependent variable.
  • \(x\) is the independent variable.
  • \(m\) is the slope of the line which represents the rate of change.
  • \(b\) is the y-intercept that shows where the line crosses the y-axis.
When there is a positive linear regression, the slope \(m\) of the line will be positive, indicating that as \(x\) increases, \(y\) increases as well. It provides a visual and mathematical way to understand how changes in one variable might predict changes in another.
Dependent Variables
A dependent variable is the variable that researchers are interested in explaining or predicting. In a linear relationship, it reflects how much it is influenced by the independent variable (or variables).
  • In our context, \(y\) is the dependent variable.
  • It changes in response to \(x\), the independent variable.
Understanding dependent variables is crucial for interpreting data models and establishing cause-and-effect connections. For example, if the dependent variable is the test score, and the independent variable is the number of hours studied, you can reasonably measure how changes in study hours affect the score. This clarity can be used in various fields, from predicting economic trends to formulating scientific research.

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

Can the values of \(B\) and \(\rho\) calculated for the same population data have different signs? Explain.

What are the degrees of freedom for a simple linear regression model?

The following table gives information on the calorie count and grams of fat for 8 of the many types of bagels produced and sold by Panera Bread. $$ \begin{array}{lcc} \hline \text { Bagel } & \text { Calories } & \text { Fat (grams) } \\ \hline \text { Asiago Cheese } & 330 & 6.0 \\ \text { Blueberry } & 340 & 1.5 \\ \text { Cinnamon Crunch } & 420 & 6.0 \\ \text { Cinnamon Swirl \& Raisin } & 320 & 2.0 \\ \text { Everything } & 300 & 2.5 \\ \text { French Toast } & 350 & 4.0 \\ \text { Plain } & 290 & 1.5 \\ \text { Sesame } & 310 & 3.0 \\ \hline \end{array} $$ a. Find the least squares regression line with calories as the dependent variable and fat content as the independent variable. b. Make a 95\% confidence interval for \(B\). c. Test at the \(5 \%\) significance level whether \(B\) is different from \(14 .\)

Explain the meaning of the words simple and linear as used in simple linear regression.

The CTO Corporation has a large number of chain restaurants throughout the United States. The research department at the company wanted to find if the restaurants' sales depend on the mean income of households in the related areas. The company collected information on these two variables for 10 restaurants randomly selected from different areas. The following table gives information on the weekly sales (in thousands of dollars) of these restaurants and the mean annual incomes (in thousands of dollars) of the households for those areas. $$ \begin{array}{l|llllllllll} \hline \text { Sales } & 26 & 38 & 23 & 30 & 22 & 40 & 44 & 32 & 28 & 47 \\ \hline \text { Income } & 46 & 63 & 48 & 52 & 32 & 55 & 58 & 49 & 41 & 72 \\ \hline \end{array} $$ a. Taking income as an independent variable and sales as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the least squares regression line. c. Briefly explain the meaning of the values of \(a\) and \(b\) calculated in part b. d Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(95 \%\) confidence interval for \(B\). g. Test at a \(2.5 \%\) significance level whether \(B\) is positive. h. Using a \(2.5 \%\) significance level, test whether \(\rho\) is positive.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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.