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Chapter 9: Problem 3

Suppose two variables are negatively correlated. Does the response variable increase or decrease as the explanatory variable increases?

Short Answer

Expert verified
The response variable decreases.

Step by step solution

01

Understanding Correlation

Correlation is a statistical measure that indicates the extent to which two variables change together. If two variables are negatively correlated, it means that as one variable increases, the other variable tends to decrease.
02

Identifying Variables

In this context, we have two variables: an explanatory variable and a response variable. The explanatory variable is the one that we assume affects or explains changes in the response variable.
03

Analyzing the Negative Correlation

Since the variables are negatively correlated, as the explanatory variable increases, the response variable will do the opposite, which is to decrease.
04

Conclusion

Based on the negative correlation between the variables, the response variable decreases as the explanatory variable increases.

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Key Concepts

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

Negative Correlation
When discussing correlation in statistics, it's important to understand what negative correlation entails. In a negative correlation, as one variable increases, the other decreases. Imagine a seesaw; when one side goes up, the other side comes down. This is exactly how negative correlation works.
For instance, if you're studying the relationship between study hours and the number of mistakes on a test, you might find a negative correlation. As more hours are spent studying, the number of mistakes usually drops.
  • A negative correlation is expressed with a correlation coefficient between -1 and 0.
  • A value closer to -1 indicates a stronger negative relationship.
Negative correlations are common in daily life, and recognizing them helps in understanding the behavior of different datasets or phenomena.
Explanatory Variable
In any statistical relationship, it's crucial to pinpoint what is causing change. This brings us to the concept of the explanatory variable. The explanatory variable, sometimes referred to as the independent variable, is the one that you manipulate or observe to see how it affects the other variable, often termed the response variable.
A good way to remember this is to think like a scientist performing an experiment:
  • The explanatory variable is similar to the condition you change or control, like adjusting the temperature to see its effect on ice melting.
  • It explains or predicts changes within the response variable.
When analyzing negative correlation, the explanatory variable's increase often leads to a decrease in the response variable. Understanding this role is foundational when attempting to infer causation from correlation.
Response Variable
Grasping the role of the response variable is another key concept in statistics. The response variable, also known as the dependent variable, changes in reaction to the explanatory variable. Think of it as the effect or outcome that you are measuring.
In experiments and analyses:
  • The response variable is what you expect to change as a result of changes made to the explanatory variable.
  • Its behavior is what you are attempting to predict or understand.
In situations where there is a negative correlation, as the explanatory variable increases, the response variable will behave in the opposite manner, often decreasing. Knowing what the response variable is helps researchers and analysts fine-tune their models and improve predictions.

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Most popular questions from this chapter

Fuming because you are stuck in traffic? Roadway congestion is a costly item, in both time wasted and fuel wasted. Let \(x\) represent the average annual hours per person spent in traffic delays and let \(y\) represent the average annual gallons of fuel wasted per person in traffic delays. A random sample of eight cities showed the following data (Reference: Statistical Abstract of the United States, 122 nd Edition). $$ \begin{array}{l|llllllll} \hline x(\mathrm{hr}) & 28 & 5 & 20 & 35 & 20 & 23 & 18 & 5 \\ \hline y(\mathrm{gal}) & 48 & 3 & 34 & 55 & 34 & 38 & 28 & 9 \\ \hline \end{array} $$ (a) Draw a scatter diagram for the data. Verify that \(\Sigma x=154, \Sigma x^{2}=3712\), \(\Sigma y=249, \Sigma y^{2}=9959\), and \(\Sigma x y=6067\). Compute \(r\) The data in part (a) represent average annual hours lost per person and average annual gallons of fuel wasted per person in traffic delays. Suppose that instead of using average data for different cities, you selected one person at random from each city and measured the annual number of hours lost \(x\) for that person and the annual gallons of fuel wasted \(y\) for the same person. $$ \begin{array}{l|cccccccc} \hline x(\mathrm{hr}) & 20 & 4 & 18 & 42 & 15 & 25 & 2 & 35 \\ \hline y(\mathrm{gal}) & 60 & 8 & 12 & 50 & 21 & 30 & 4 & 70 \\ \hline \end{array} $$ (b) Compute \(\bar{x}\) and \(\bar{y}\) for both sets of data pairs and compare the averages. Compute the sample standard deviations \(s_{x}\) and \(s_{y}\) for both sets of data pairs and compare the standard deviations. In which set are the standard deviations for \(x\) and \(y\) larger? Look at the defining formula for \(r\), Equation \(1 .\) Why do smaller standard deviations \(s_{x}\) and \(s_{y}\) tend to increase the value of \(r\) ? (c) Make a scatter diagram for the second set of data pairs. Verify that \(\Sigma x=161, \quad \Sigma x^{2}=4583, \quad \Sigma y=255, \quad \Sigma y^{2}=12,565\), and \(\Sigma x y=7071 .\) Compute \(r\). (d) Compare \(r\) from part (a) with \(r\) from part (c). Do the data for averages have a higher correlation coefficient than the data for individual measurements? List some reasons why you think hours lost per individual and fuel wasted per individual might vary more than the same quantities averaged over all the people in a city.

Trevor conducted a study and found that the correlation between the price of a gallon of gasoline and gasoline consumption has a linear correlation coefficient of \(-0.7 .\) What does this result say about the relationship between price of gasoline and consumption? The study included gasoline prices ranging from \(\$ 2.70\) to \(\$ 5.30\) per gallon. Is it reliable to apply the results of this study to prices of gasoline higher than \(\$ 5.30\) per gallon? Explain.

When we use a least-squares line to predict \(y\) values for \(x\) values beyond the range of \(x\) values found in the data, are we extrapolating or interpolating? Are there any concerns about such predictions?

The initial visual impact of a scatter diagram depends on the scales used on the \(x\) and \(y\) axes. Consider the following data: $$ \begin{array}{l|llllll} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 1 & 4 & 6 & 3 & 6 & 7 \\ \hline \end{array} $$ (a) Make a scatter diagram using the same scale on both the \(x\) and \(y\) axes (i.e., make sure the unit lengths on the two axes are equal). (b) Make a scatter diagram using a scale on the \(y\) axis that is twice as long as that on the \(x\) axis. (c) Make a scatter diagram using a scale on the \(y\) axis that is half as long as that on the \(x\) axis. (d) On each of the three graphs, draw the straight line that you think best fits the data points. How do the slopes (or directions) of the three lines appear to change? Note: The actual slopes will be the same; they just appear different because of the choice of scale factors.

Is the magnitude of an earthquake related to the depth below the surface at which the quake occurs? Let \(x\) be the magnitude of an earthquake (on the Richter scale), and let \(y\) be the depth (in kilometers) of the quake below the surface at the epicenter. The following is based on information taken from the National Earthquake Information Service of the U.S. Geological Survey. Additional data may be found by visiting the Brase/Brase statistics site at $$ \begin{array}{l|lllllll} \hline x & 2.9 & 4.2 & 3.3 & 4.5 & 2.6 & 3.2 & 3.4 \\ \hline y & 5.0 & 10.0 & 11.2 & 10.0 & 7.9 & 3.9 & 5.5 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=24.1, \Sigma x^{2}=85.75, \Sigma y=53.5\), \(\Sigma y^{2}=458.31\), and \(\Sigma x y=190.18\). Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.
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