/*! 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 29 How would you interpret the find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

How would you interpret the findings of a correlation study that reported a linear correlation coefficient of \(+0.37 ?\)

Short Answer

Expert verified
A linear correlation coefficient of \(+0.37\) shows a weak positive correlation between the variables studied. This signifies that as one variable increases, the other also tends to increase. However, the relationship isn't very strong and doesn't imply causation.

Step by step solution

01

Understanding Correlation Coefficient

A correlation coefficient is a measure that determines the degree to which two variables' movements are associated. The most common measure is Pearson's Correlation Coefficient, which gives values between -1 and +1. The closer the coefficient to +1 or -1 the stronger the correlation, either positive or negative respectively. A positive value denotes a direct correlation (one increases as the other does), while a negative value indicates an inverse correlation (one increases while the other decreases).
02

Interpreting the Coefficient

Given the coefficient of \(+0.37\), it implies that there is a positive but weak correlation between the two variables under consideration. This means that as one variable increases, the other variable tends to increase as well, but the relationship isn't very strong.
03

Limitations and Further Analysis

It's important to mention that a correlation doesn't imply causation. In addition, a weak correlation suggests that there may be other factors at play influencing the variables. Therefore, further detailed analysis might be necessary to fully understand the 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.

Pearson's Correlation
Pearson's correlation coefficient is a statistical measure used to evaluate the strength and direction of the linear relationship between two quantitative variables. It ranges from -1 to +1.
This range helps identify how closely the data fit a linear trend line. A fundamental aspect of Pearson's correlation is its simplicity and wide application. Here are some key points about it:
  • Values close to +1 imply a strong positive linear relationship.
  • Values near 0 suggest no linear relationship whatsoever.
  • Negative values close to -1 indicate a strong negative linear relationship.
Pearson's correlation makes assumptions about data normality and homoscedasticity, which means the data should ideally be distributed normally and have constant variance along the trend line.
Positive Correlation
A positive correlation means that as one variable increases, the other variable also increases. The degree of this relationship is determined by the correlation coefficient value.
In our example, the correlation coefficient observed is +0.37. This indicates a positive correlation, where both variables tend to increase together.
However, it's important to note certain aspects of positive correlation with respect to its magnitude:
  • +0.3 to +0.5 describes a weak positive correlation.
  • +0.5 to +0.7 implies a moderate positive correlation.
  • +0.7 and above indicates a strong positive correlation.
Thus, even if a correlation is positive like in this case, at +0.37, it's still weak and suggests that many other unobserved variables might be influencing the dependent variable's behavior.
Interpretation of Correlation
Interpreting correlation requires understanding that it only describes the extent and direction of a relationship, without implying causation.
A linear correlation coefficient of +0.37, as in the exercise, suggests a weak positive linkage, meaning the variables slightly tend to move in the same direction together.
Here are some critical aspects to consider when interpreting correlation coefficients:
  • Correlation does not imply causation; it merely indicates association.
  • External factors not included in the model may influence the variables.
  • Correlation should be considered a tool for preliminary examination rather than definitive proof of linkage.
Thus, when you come across any correlation value, especially one like +0.37, it is crucial to remember its limitations and explore further if accurate predictions or understandings are required.

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

A study was conducted to investigate the relationship between the resale price, \(y\) eds of dollars),and the age, \(x\) (in years), of midsize luxury American automobiles. The equation of the line of best fit was determined to be \(\hat{y}=185.7-21.52 x\) a. Find the resale value of such a car when it is 3 years old. b. Find the resale value of such a car when it is 6 years old. c. What is the average annual decrease in the resale price of these cars?

A.J. used linear regression to help him understand his monthly telephone bill. The line of best fit was \(\hat{y}=23.65+1.28 x,\) where \(x\) is the number of long-distance calls made during a month, and \(y\) is the total telephone cost for a month. In terms of number of long-distance calls and cost: a. Explain the meaning of the \(y\) -intercept, 23.65 b. Explain the meaning of the slope, 1.28

The June 2009 unemployment rates for eastern and western U.S. states were as follows:$$\begin{array}{lcccccc}\hline \text { Eastern } & 8.0 & 10.6 & 10.1 & 7.3 & 9.2 & 11.0 & 12.1 & 7.2 \\\\\text { Western } & 8.7 & 11.6 & 8.4 & 6.4 & 12.0 & 12.2 & 5.7 & 9.3 \\\\\hline\end{array}$$.Display these rates as two dotplots using the same scale; compare means and medians.

Players, teams, and fans are interested in seeing their leading scorers score lots of points, yet at the same time the number of personal fouls they commit tends to limit their playing time. For the leading scorer on each team, the table lists the number of minutes per game, MPG, and the number of personal fouls committed per game, PFPG, during the \(2008 / 2009\) NBA season.$$\begin{array}{lll|lll} \text { Team } & \text { MPG } & \text { PFPG } & \text { Team } & \text { MPG } & \text { PFPG } \\ \hline \text { Hawks } & 39.6 & 2.23 & \text { Bucks } & 36.4 & 1.36 \\ \text { Celics } & 37.5 & 2.65 & \text { Timberwolves } & 36.7 & 2.82 \\ \text { Hornets } & 37.6 & 2.96 & \text { Nets } & 36.1 & 2.38 \\ \text { Bulls } & 36.6 & 2.24 & \text { Hornets } & 38.5 & 2.72 \\ \text { Cavaliers } & 37.7 & 1.72 & \text { Knicks } & 29.8 & 2.78 \\ \text { Mavericks } & 37.7 & 2.17 & \text { Thunder } & 39.0 & 1.81 \\ \text { Nuggels } & 34.5 & 2.95 & \text { Magic } & 35.7 & 3.42 \\ \text { Pistons } & 34.0 & 2.63 & \text { 76ers } & 39.9 & 1.85 \end{array}$$.a. Construct a scatter diagram. b. Describe the pattern displayed. Are there any unusual characteristics displayed? c. Calculate the correlation coefficient. d. Does the value of the correlation coefficient seem reasonable?

Phi \((\Phi=1.618033988749895 \ldots),\) most often pronounced "fi" (like "fly"), is simply an irrational number like pi \((\pi=3.14159265358979 \ldots),\) but one with many unusual mathematical properties. Phi is the basis for the Golden Ratio. (Visit http://goldennumber.net/ to learn other interesting things about phi.) a. If every person's arm displayed the exact Golden Ratio, describe the appearance of a scatter diagram where the length of the forearm, \(y,\) and the length of the hand, \(x,\) have been plotted. b. since body proportions vary from person to person, describe the appearance of a scatter diagram where the length of the forearm, \(y\), and the length of the hand, \(x,\) have been plotted for 25 people whose two lengths had been measured.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.