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Assignment 2: Linear Correlation and...

Assignment 2: Linear Correlation and Hypothesis Testing

You will be examining pairs of variables to determine if there is any linear relation between the variables.

Background Reading:

Linear relations are of interest to researchers, as they are easy to describe and are useful for making predictions. The most frequently used measure of linear correlation is the Pearson correlation coefficient, R2.

The Pearson correlation coefficient measures how closely the data points are to a best-fit straight line. More precisely, a computer will find the residuals (the distance of each data point to the line), square them, and attempt to minimize the residuals. It can also be shown that R^2 measures the fraction of variation in y that can be attributed to variation in x. (Likewise, R^2 also represents the fraction of variation in x that can be attributed to variation in y).

Values of R^2 close to 1 indicate a strong linear relation between the variables; values of R^2 close to 0 indicate little or no linear relation.

A linear relation between variables need not mean that one variable causes the behavior in the other. Also, an R^2 value may not detect non-linear relationships that might exist between two variables.

An R or R^2 value can be subjected to hypothesis testing to determine if it is statistically significant, with the R^2 value used as the test statistic. It is necessary to specify the confidence interval, the degrees of freedom (equal to the sample size minus 1), and whether the confidence interval is two tailed or one tailed.

The researcher is usually testing a claim that R or R^2 is at least equal to some stated value, so a one-tailed test would be used.

Notice that R is found as the square root of R^2. The data are then plotted on a scatterplot. If the scatterplot shows that the x and y variables increase together, R will have a positive sign. If they move in opposite directions (i.e., y decreases when x increases), R will have a negative sign.

Sample: As an example of observing possible linear correlations, please open and examine Worksheet 1. This data shows the number of hours a student spent preparing for an exam vs. the score out of 100 earned on that exam.

Directions:

In a Word document, include the following:

Please report the R^2 linear correlation coefficient and the linear regression equation produced in Excel.

Would the correlation value R be positive or negative?

What is the implication of any linear relation found between hours spent studying and the exam score earned?

You will be using the instructions in the Excel Linear Correlation and Regression document.

Submit your assignment to the M3: Assignment 2 Dropbox by Wednesday, October 31, 2012.

Complete scatterplot and attach as an Excel file. The fraction of variation in one variable should be accounted for by variation of the other.

Report the R^2 correlation coefficient and linear regression equation with slope and intercept included and stated whether the value of R is positive or negative.

Explain the implication of any linear relation, including its three components (scatterplot, R^2 value and linear equation) found between hours spent studying, and the exam score earned.

You will be examining pairs of variables to determine if there is any linear relation between the variables.

Background Reading:

Linear relations are of interest to researchers, as they are easy to describe and are useful for making predictions. The most frequently used measure of linear correlation is the Pearson correlation coefficient, R2.

The Pearson correlation coefficient measures how closely the data points are to a best-fit straight line. More precisely, a computer will find the residuals (the distance of each data point to the line), square them, and attempt to minimize the residuals. It can also be shown that R^2 measures the fraction of variation in y that can be attributed to variation in x. (Likewise, R^2 also represents the fraction of variation in x that can be attributed to variation in y).

Values of R^2 close to 1 indicate a strong linear relation between the variables; values of R^2 close to 0 indicate little or no linear relation.

A linear relation between variables need not mean that one variable causes the behavior in the other. Also, an R^2 value may not detect non-linear relationships that might exist between two variables.

An R or R^2 value can be subjected to hypothesis testing to determine if it is statistically significant, with the R^2 value used as the test statistic. It is necessary to specify the confidence interval, the degrees of freedom (equal to the sample size minus 1), and whether the confidence interval is two tailed or one tailed.

The researcher is usually testing a claim that R or R^2 is at least equal to some stated value, so a one-tailed test would be used.

Notice that R is found as the square root of R^2. The data are then plotted on a scatterplot. If the scatterplot shows that the x and y variables increase together, R will have a positive sign. If they move in opposite directions (i.e., y decreases when x increases), R will have a negative sign.

Sample: As an example of observing possible linear correlations, please open and examine Worksheet 1. This data shows the number of hours a student spent preparing for an exam vs. the score out of 100 earned on that exam.

Directions:

In a Word document, include the following:

Please report the R^2 linear correlation coefficient and the linear regression equation produced in Excel.

Would the correlation value R be positive or negative?

What is the implication of any linear relation found between hours spent studying and the exam score earned?

You will be using the instructions in the Excel Linear Correlation and Regression document.

Submit your assignment to the M3: Assignment 2 Dropbox by Wednesday, October 31, 2012.

Complete scatterplot and attach as an Excel file. The fraction of variation in one variable should be accounted for by variation of the other.

Report the R^2 correlation coefficient and linear regression equation with slope and intercept included and stated whether the value of R is positive or negative.

Explain the implication of any linear relation, including its three components (scatterplot, R^2 value and linear equation) found between hours spent studying, and the exam score earned.

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