/*! 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.

Short Answer

Expert verified
In a positive linear relationship, if \(x\) increases, \(y\) will also increase.

Step by step solution

01

Understanding the Concept of Positive Linear Relationship

In a positive linear relationship, the two variables \(x\) and \(y\) change in the same direction. This means that if one variable increases, the other will also increase, and if one variable decreases, the other will also decrease. This happens at a constant rate, which is the slope of the line that represents the relationship.
02

Relation between the Increase of \(x\) and \(y\)

In the case of a positive linear relationship, if \(x\) increases, the value of \(y\) will also increase. This is because both variables change in the same direction, as established in the concept of a positive linear relationship.

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 expresses the extent to which two variables are linearly related. It is denoted by the symbol \( r \). The value of \( r \) ranges from -1 to 1.
  • A correlation of 1 indicates a perfect positive linear relationship, meaning that as one variable increases, the other also increases at a constant rate.
  • A correlation of -1 indicates a perfect negative linear relationship, which means as one variable increases, the other decreases consistently.
  • An \( r \) value of 0 indicates no linear relationship between the variables.
In the context of our problem, a positive linear relationship reveals a correlation greater than zero but not exceeding 1. This suggests that as the variable \( x \) increases, the variable \( y \) increases as well, demonstrating a direct relationship. Understanding correlation helps in forecasting and making predictions about data trends.
Slope
Slope is an essential concept when discussing linear relationships. It represents the rate at which one variable changes in relation to another. In a linear equation, the slope is usually represented by \( m \) in the equation \( y = mx + c \), where \( m \) tells us how much \( y \) changes for a unit change in \( x \).
  • If the slope \( m \) is positive, it confirms a positive linear relationship, indicating that as \( x \) increases, \( y \) does too.
  • The steepness of the slope also conveys how quickly \( y \) increases with respect to \( x \). A steeper slope means a larger change in \( y \) for a small change in \( x \).
In the given exercise, a positive slope signifies that as \( x \) grows, \( y \) simultaneously grows. Thus, the direction and magnitude of the slope are crucial in understanding the dynamics of linear relationships.
Linear Regression
Linear regression is a statistical technique used to model and analyze the relationships between variables. It's primarily employed to understand how the dependent variable \( y \) changes with respect to the independent variable \( x \). The equation of a simple linear regression is generally written as \( y = mx + c \), where \( m \) represents the slope, and \( c \) is the y-intercept.
  • In regression analysis, the goal is to find the best-fitting line through the data points that minimizes the distance between the line and all points.
  • A positive slope in the regression line indicates a positive linear relationship, which aligns with our problem where both variables increase together.
Linear regression not only reveals the relationship between variables but also forecasts future values, thus serving as a powerful predictive tool in statistics.
Variables in Statistics
Variables are foundational in statistics, representing elements that can vary or change within data sets and analyses. In our exercise, two types of variables are typically involved: the independent variable and the dependent variable.
  • The **independent variable** (often denoted as \( x \)) is the one that is manipulated or controlled in an experiment to observe its effect on another variable.
  • The **dependent variable** (often denoted as \( y \)) is the variable being tested and measured, affected by changes in the independent variable.
Understanding variables is crucial for any statistical analysis as they help identify cause-effect relationships and allow us to make data-driven conclusions. In the exercise provided, \( x \) is the independent variable impacting \( y \), demonstrating a synchronous change as they both increase.

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 owner of a small factory that produces working gloves is concerned about the high cost of air conditioning in the summer but is afraid that keeping the temperature in the factory too high will lower productivity. During the summer, he experiments with temperature settings from \(68^{\circ} \mathrm{F}\) to \(81^{\circ} \mathrm{F}\) and measures each day's productivity. The following table gives the temperature and the number of pairs of gloves (in hundreds) produced on each of the 8 randomly selected days. $$ \begin{array}{l|cccccccc} \hline \text { Temperature }\left({ }^{\circ} \mathrm{F}\right) & 72 & 71 & 78 & 75 & 81 & 77 & 68 & 76 \\ \hline \text { Pairs of gloves } & 37 & 37 & 32 & 36 & 33 & 35 & 39 & 34 \\ \hline \end{array} $$ a. Do the pairs of gloves produced depend on temperature, or does temperature depend on pairs of gloves produced? Do you expect a positive or a negative relationship between these two variables? b. Taking temperature as an independent variable and pairs of gloves produced as a dependent variable, compute \(\mathrm{SS}_{x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x v}\) c. Find the least squares regression line. d. Interpret the meaning of the values of \(a\) and \(b\) calculated in part \(\mathrm{c}\). e. Plot the scatter diagram and the regression line. f. Calculate \(r\) and \(r^{2}\), and explain what they mean. g. Compute the standard deviation of errors. h. Predict the number of pairs of gloves produced when \(x=74\). i. Construct a \(99 \%\) confidence interval for \(B\). j. Test at the \(5 \%\) significance level whether \(B\) is negative. \(\mathrm{k}_{4}\) Using \(\alpha=.01\) can you conclude that \(\rho\) is negative?

For a sample data set, the slope \(b\) of the regression line has a negative value. Which of the following is true about the linear correlation coefficient \(r\) calculated for the same sample data? a. The value of \(r\) will be positive. b. The value of \(r\) will be negative. c. The value of \(r\) can be positive or negative.

The management of a supermarket wants to find if there is a relationship between the number of times a specific product is promoted on the intercom system in the store and the number of units of that product sold. To experiment, the management selected a product and promoted it on the intercom system for 7 days. The following table gives the number of times this product was promoted each day and the number of units sold. $$ \begin{array}{cc} \hline \begin{array}{c} \text { Number of Promotions } \\ \text { per Day } \end{array} & \begin{array}{c} \text { Number of Units Sold } \\ \text { per Day (hundreds) } \end{array} \\ \hline 15 & 11 \\ 22 & 22 \\ 42 & 30 \\ 30 & 26 \\ 18 & 17 \\ 12 & 15 \\ 38 & 23 \\ \hline \end{array} $$ a. With the number of promotions as an independent variable and the number of units sold as a dependent variable, what do you expect the sign of \(B\) in the regression line \(y=A+B x+\epsilon\) will be? b. Find the least squares regression line \(\hat{y}=a+b x\). Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Compute \(r\) and \(r^{2}\) and explain what they mean. e. Predict the number of units of this product sold on a day with 35 promotions. f. Compute the standard deviation of errors. g. Construct a \(98 \%\) confidence interval for \(B\). h. Testing at the \(1 \%\) significance level, can you conclude that \(B\) is positive? i. Using \(\alpha=.02\), can you conclude that the correlation coefficient is different from zero?

Refer to Exercise \(13.25\). The data on ages (in years) and prices (in hundreds of dollars) for eight cars of a specific model are reproduced from that exercise. $$ \begin{array}{l|rrrrrrrr} \hline \text { Age } & 8 & 3 & 6 & 9 & 2 & 5 & 6 & 3 \\ \hline \text { Price } & 45 & 210 & 100 & 33 & 267 & 134 & 109 & 235 \\ \hline \end{array} $$ a. Do you expect the ages and prices of cars to be positively or negatively related? Explain. b. Calculate the linear correlation coefficient. c. Test at the \(2.5 \%\) significance level whether \(\rho\) is negative.

Briefly explain the difference between a deterministic and a probabilistic regression model.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.