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

Short Answer

Expert verified
As the explanatory variable increases, the response variable also increases.

Step by step solution

01

Understand Positive Correlation

A positive correlation between two variables means that as one increases, the other also tends to increase. Both variables move in the same direction.
02

Identify Variables

Recognize the explanatory variable, which is the one you manipulate or observe, and the response variable, which is the outcome you measure as the explanatory variable changes.
03

Apply the Concept of Positive Correlation

Since the question states that the two variables are positively correlated, as the explanatory variable increases, the response variable also increases.

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Key Concepts

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

Understanding the Response Variable
In any study or experiment involving variables, the response variable is what researchers are primarily interested in. It's the main factor they want to observe and measure. You can think of it as the effect in a cause-and-effect relationship. For example, if a study is analyzing the impact of study time on test scores, the response variable would be the test scores because that's what changes are being measured.

To identify the response variable, ask yourself what the result of the study's manipulation is. This variable reacts to changes made to another variable, the explanatory variable. Sometimes, it's also called the dependent variable, to highlight its reliance on other factors in the experiment.
Identifying the Explanatory Variable
The explanatory variable plays a crucial role in determining the outcomes explored in a study. It's the element that researchers control or choose to observe to see how it impacts the response variable. Often called the independent variable, it acts like the 'cause' in a cause-and-effect scenario.

In our example of studying the relationship between study time and test scores, the amount of time spent studying is the explanatory variable. Researchers manipulate it to see the effect it causes on test scores. The explanatory variable helps set up the framework of an experiment or observation, ensuring that any changes in the response variable can be linked back to variations in this independent variable.
  • This variable does not depend on other variables in the experiment's context.
  • It helps us understand the relationship and how one factor influences another.
Mastering the Step-by-Step Solution
Approaching problems with a step-by-step method is an effective way to gain clarity and avoid overlooking important details. Let’s break down the process like in the original solution to understanding how positive correlation affects variables:

  • Understand Positive Correlation: Know that in a positive relationship, both variables move in the same direction. If one rises, the other also tends to rise.
  • Identify the Variables: Determine which is your explanatory variable (cause) and which is your response variable (effect).
  • Apply the Concept: With positive correlation identified, as you increase the explanatory variable, watch as the response variable also increases.

Utilizing a step-by-step approach not only ensures thorough understanding but also makes the solution more manageable. This technique provides a structured way to approach similar problems in the future, empowering you with a solid problem-solving foundation.

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Most popular questions from this chapter

What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let \(x=\) depth of dive in meters, and let \(y=\) optimal time in hours. A random sample of divers gave the following data (based on information taken from Medical Physiology by A. C. Guyton, M.D.). $$ \begin{array}{l|lllllll} \hline x & 14.1 & 24.3 & 30.2 & 38.3 & 51.3 & 20.5 & 22.7 \\ \hline y & 2.58 & 2.08 & 1.58 & 1.03 & 0.75 & 2.38 & 2.20 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=201.4, \Sigma y=12.6, \Sigma x^{2}=6735.46, \Sigma y^{2}=25.607, \Sigma x y=\) 311.292, and \(r \approx-0.976\). (b) Use a \(1 \%\) level of significance to test the claim that \(\rho<0\). (c) Verify that \(S_{e} \approx 0.1660, a \approx 3.366\), and \(b \approx-0.0544\). (d) Find the predicted optimal time in hours for a dive depth of \(x=18\) meters. (e) Find an \(80 \%\) confidence interval for \(y\) when \(x=18\) meters. (f) Use a \(1 \%\) level of significance to test the claim that \(\beta<0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and interpret its meaning.

Can a low barometer reading be used to predict maximum wind speed of an approaching tropical cyclone? Data for this problem are based on information taken from Weatherwise \((\) Vol. 46, No. 1\()\), a publication of the American Meteorological Society. For a random sample of tropical cyclones, let \(x\) be the lowest pressure (in millibars) as a cyclone approaches, and let \(y\) be the maximum wind speed (in miles per hour) of the cyclone. $$ \begin{array}{l|rrrrrr} \hline x & 1004 & 975 & 992 & 935 & 985 & 932 \\ \hline y & 40 & 100 & 65 & 145 & 80 & 150 \\ \hline \end{array} $$ (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 \(\Sigma x=5823, \Sigma x^{2}=5,655,779, \Sigma y=580\), \(\Sigma y^{2}=65,750\), and \(\Sigma x y=556,315 .\) Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

Please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums \(\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2}\), and \(\sum x y\) and the value of the sample correlation coefficient \(\underline{r}\) (c) Find \(\bar{x}, \bar{y}, a\), and \(b .\) Then find the equation of the least- squares line \(\hat{y}=a+b x\) (d) Graph the least-squares line on your scatter diagram. Be sure to use the point \((\bar{x}, \bar{y})\) as one of the points on the line. (e) Find the value of the coefficient of determination \(r^{2} .\) What percentage of the variation in \(y\) can be explained by the corresponding variation in \(x\) and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. An economist is studying the job market in Denver area neighborhoods. Let \(x\) represent the total number of jobs in a given neighborhood, and let \(y\) represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs). $$ \begin{array}{l|rrrrrr} \hline x & 16 & 33 & 50 & 28 & 50 & 25 \\ \hline y & 2 & 3 & 6 & 5 & 9 & 3 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=202, \Sigma y=28, \Sigma x^{2}=7754\), \(\Sigma y^{2}=164, \Sigma x y=1096\), and \(r \approx 0.860\) (f) For a neighborhood with \(x=40\) jobs, how many are predicted to be entrylevel jobs?

For a fixed confidence level, how does the length of the confidence interval for predicted values of \(y\) change as the corresponding \(x\) values become farther away from \(\bar{x}\) ?

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

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