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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.

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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.

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Most popular questions from this chapter

Explain each of the following concepts. You may use graphs to illustrate each concept. A. Perfect positive linear correlation b. Perfect negative linear correlation c. Strong positive linear correlation d. Strong negative linear correlation e. Weak positive linear correlation f. Weak negative linear correlation g. No linear correlation

A diabetic is interested in determining how the amount of aerobic exercise impacts his blood sugar. When his blood sugar reaches \(170 \mathrm{mg} / \mathrm{dL}\), he goes out for a run at a pace of 10 minutes per mile. On different days, he runs different distances and measures his blood sugar after completing his run. Note: The preferred blood sugar level is in the range of 80 to \(120 \mathrm{mg} / \mathrm{dL}\). Levels that are too low or too high are extremely dangerous. The data generated are given in the following table. $$ \begin{array}{l|rrrrrrrrrrrr} \hline \text { Distance (miles) } & 2 & 2 & 2.5 & 2.5 & 3 & 3 & 3.5 & 3.5 & 4 & 4 & 4.5 & 4.5 \\ \hline \text { Blood sugar (mg/dL) } & 136 & 146 & 131 & 125 & 120 & 116 & 104 & 95 & 85 & 94 & 83 & 75 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between distance run and blood sugar level? b. Find the predictive regression equation of blood sugar level on the distance run. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\underline{b}\). d. Plot the predictive regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the predictive regression line e. Calculate the predicted blood sugar level count after a run of \(3.1\) miles \((5\) kilometers) f. Estimate the blood sugar level after a 10 -mile run. Comment on this finding.

A car rental company charges \(\$ 50\) a day and 20 cents per mile for renting a car. Let \(y\) be the total rental charges (in dollars) for a car for one day and \(x\) be the miles driven. The equation for the relationship between \(x\) and \(y\) is $$ y=50+.20 x $$ a. How much will a person pay who rents a car for one day and drives it 100 miles? b. Suppose each of 20 persons rents a car from this agency for one day and drives it 100 miles. Will each of them pay the same amount for renting a car for a day or do you expect each person to pay a different amount? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?

Explain the meaning and concept of SSE. You may use a graph for illustration purposes.

The recommended air pressure in a basketball is between 7 and 9 pounds per square inch (psi). When dropped from a height of 6 feet, a properly inflated basketball should bounce upward between 52 and 56 inches. The basketball coach at a local high school purchased 10 new basketballs for the upcoming season, inflated the balls to pressures between 7 and 9 psi, and performed the bounce test mentioned above. The data obtained are given in the following table. $$ \begin{array}{l|rrrrrrrrrr} \hline \text { Pressure (psi) } & 7.8 & 8.1 & 8.3 & 7.4 & 8.9 & 7.2 & 8.6 & 7.5 & 8.1 & 8.5 \\ \hline \text { Bounce height (inches) } & 54.1 & 54.3 & 55.2 & 53.3 & 55.4 & 52.2 & 55.7 & 54.6 & 54.8 & 55.3 \\ \hline \end{array} $$ a. With the pressure as an independent variable and bounce height 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. Interpret the meaning of the values of \(a\) and \(b\) calculated in part b. d. Calculate \(r\) and \(r^{2}\) and explain what they mean. e. Compute the standard deviation of errors. f. Predict the bounce height of a basketball for \(x=8.0\). g. Construct a \(98 \%\) confidence interval for \(B\). h. Test at the \(5 \%\) significance level whether \(B\) is different from zero. i. Using \(\alpha=.05\), can you conclude that \(\rho\) is different from zero?
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