PSY 520 Week 3 Discussion 1: Linear Correlations

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PSY 520 Week 3 Discussion 1: Linear Correlations

PSY 520 Week 3 Discussion 1: Linear Correlations

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Review the video Linear Correlation in the Calculations section of the Statistics Visual Learner media piece.

It is sometimes said that the higher the correlation between two variables, the more likely the relationship is causal. Do you think this is correct Discuss? PSY 520 Week 3 Discussion 1: Linear Correlations

Details:

Use the document “Probability Project” to complete the assignment.

While APA format is not required for the body of this assignment, solid academic writing is expected, and documentation of sources should be presented using APA formatting guidelines, which can be found in the APA Style Guide, located in the Student Success Center. PSY 520 Week 3 Discussion 1: Linear Correlations

Topic 3 – Probability Project

Directions

Use the following information to complete the assignment. While APA format is not required for the body of this assignment, solid academic writing is expected, and documentation of sources should be presented using APA formatting guidelines, which can be found in the APA Style Guide, located in the Student Success Center. PSY 520 Week 3 Discussion 1: Linear Correlations

There are many misconceptions about probability which may include the following.

  • All events are equally likely
  • Later events may be affected by or compensate for earlier ones

    PSY 520 Week 3 Discussion 1 Linear Correlations
    PSY 520 Week 3 Discussion 1 Linear Correlations
  • When determining probability from statistical data, sample size is irrelevant
  • Results of games of skill are unaffected by the nature of the participants
  • “LuckyUnlucky” numbers can influence random events PSY 520 Week 3 Discussion 1: Linear Correlations
  • In random event involving selection, results are dependent on number rather than rations
  • If events are random then the results of a series of independent events are equally likely

Click here to ORDER an A++ paper from our Verified MASTERS and DOCTORATE WRITERS: PSY 520 Week 3 Discussion 1: Linear Correlations 

The following statements are all incorrect. Explain the statements and the errors fully using the probability rules discussed in topic two.

1. I have flipped and unbiased coin three times and got heads, it is more likely to get tails the next time I flip it. PSY 520 Week 3 Discussion 1: Linear Correlations

2. The Rovers play Mustangs. The Rovers can win, loose, or draw, so the probability that they win is 13. PSY 520 Week 3 Discussion 1: Linear Correlations

3. I roll two dice and ad the results. The probability of getting a total of 6 is 112 because there are 12 different possibilities and 6 is one of them.

4. Mr. Purple has to have a major operation. 90% of the people who have this operation make a complete recovery. There is a 90% chance that Mr. Purple will make a complete recovery if he has this operation.

5. I flip two coins. The probability of getting heads and tails is 13 because I can get Heads and Heads, Heads and Tails, or Tails and Tails.

6. 13 is an unlucky number so you are less likely to win raffles with ticket number 13 than with a different dumber.

PSY 520 Week 3 Discussion 2 Latest-GCU

Discuss the strengths and weaknesses of correlational and regression studies; discuss concepts such as positive and negative correlations, correlation coefficients, confounding, and causality.

PSY 520 Week 3 Exercise Latest-GCU

Details

Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

Show all relevant work; use the equation editor in Microsoft Word when necessary.

1. Chapter 6, numbers 6.7, 6.10, and 6.11

2. Chapter 7, numbers 7.8, 7.10, and 7.13

 

Linear correlation

by Marco Taboga, PhD

Linear correlation is a measure of dependence between two random variables that can take values between -1 and 1. It is proportional to covariance and its interpretation is very similar to that of covariance. PSY 520 Week 3 Discussion 1: Linear Correlations

Table of Contents

Definition

Let X and Y be two random variables. The linear correlation coefficient (or Pearson’s correlation coefficient) between X and Y, denoted by [eq1] or by $ ho _{XY}$, is defined as follows:[eq2]where [eq3] is the covariance between X and Y and [eq4] and [eq5] are the standard deviations of X and Y.

The linear correlation coefficient is well-defined only as long as [eq6], [eq7] and [eq8] exist and are well-defined.

Note that, in principle, the ratio is well-defined only if [eq9] and [eq5] are strictly greater than zero. However, it is often assumed that [eq11] when one of the two standard deviations is zero. This is equivalent to assuming that [eq12] because [eq13] when one of the two standard deviations is zero.

Interpretation

The interpretation is similar to the interpretation of covariance: the correlation between X and Y provides a measure of how similar their deviations from the respective means are (see the lecture entitled Covariance for a detailed explanation). PSY 520 Week 3 Discussion 1: Linear Correlations

Linear correlation has the property of being bounded between $-1$ and 1:[eq14]

Thanks to this property, correlation allows us to easily understand the intensity of the linear dependence between two random variables: the closer correlation is to 1, the stronger the positive linear dependence between $X $ and Y is (and the closer it is to $-1$, the stronger the negative linear dependence between X and Y is).

Terminology

The following terminology is often used:

  1. If [eq15] then X and Y are said to be positively linearly correlated (or simply positively correlated).
  2. If [eq16] then X and Y are said to be negatively linearly correlated (or simply negatively correlated).
  3. If [eq17] then X and Y are said to be linearly correlated (or simply correlated).
  4. If [eq18] then X and Y are said to be uncorrelated. Also note that [eq19] $=0$, therefore two random variables X and Y are uncorrelated whenever [eq20].

Example

The following example shows how to compute the coefficient of linear correlation between two discrete random variables. PSY 520 Week 3 Discussion 1: Linear Correlations

Example Let X be a $2$-dimensional random vector and denote its components by X_1 and X_2. Let the support of X be[eq21]and its joint probability mass function be[eq22]The support of X_1 is[eq23]and its probability mass function is[eq24]The expected value of X_1 is[eq25]The expected value of $X_{1}^{2}$ is[eq26]The variance of X_1 is[eq27]The standard deviation of X_1 is:[eq28]The support of X_2 is:[eq29]and its probability mass function is[eq30]The expected value of X_2 is[eq31]The expected value of $X_{2}^{2}$ is[eq32]The variance of X_2 is[eq33]The standard deviation of X_2 is[eq34]Using the transformation theorem, we can compute the expected value of $X_{1}X_{2}$:[eq35]Hence, the covariance between X_1 and X_2 is[eq36]and the linear correlation coefficient is:[eq37] PSY 520 Week 3 Discussion 1: Linear Correlations