CORRELATIONAL ANALYSIS
What is a Positive Correlation?
A Positive Correlation exists if high values in one variable are associated with high values in another variable. Similarly, low values in one variable are associated with low values in the other variable.
What is a Scatter plot?
A Scatter plot is a graphical way of showing the correlation between two variables. The horizontal axis represents one variable and the vertical axis represents the other variable.
What is Zero/No Correlation?
A Zero/No Correlation exists when high values in one variable are associated to either high or low values in another variable.
What does an r value of -1 mean?
An r value of -1 represents a perfect negative correlation between two variables.
What does an r value of 0 mean?
An r value of 0 represents zero or no correlation between two variables.
What does an r value of 1 mean?
An r value of 1 represents a perfect positive correlation between two variables.
When should you not use the regression equation to make predictions?
If there is no significant linear correlation between the variables, do not use the regression equation to make predictions.
What is the range of the correlation coefficient?
The correlation coefficient ranges from +1 to -1, where zero means no correlation and 1 means strong (perfect) correlation.
What is the range of r values for moderately high correlation?
The range of r values for moderately high correlation is from +0.60 to +0.79.
What is the range of r values for negligible correlation?
The range of r values for negligible correlation is from +0.10 to +0.19.
Null Hypothesis (Ho) for Correlation Coefficient
Ho: There is no significant relationship between the two variables In other words, the correlation coefficient is equal to zero
What is a Negative Correlation?
A Negative Correlation exists if high values in one variable are associated with low values in another variable. Similarly, low values in one variable are associated with high values in the other variable.
What is a Perfect Correlation?
A Perfect Correlation exists when all the points on the scatter plot lie on a straight line.
Critical Values (CV) for Correlation Coefficient
CV is based on the level of significance, degrees of freedom, and whether the test is one-tailed or two-tailed It is used to compare with the test statistic to make a decision on whether to reject or fail to reject the null hypothesis
What is Correlation Analysis?
Correlation Analysis is the process or procedure of describing the relationship between two variables.
Formula for Computing Test Statistic for Correlation Coefficient
Formula: t = r / (sqrt((1-r^2)/(n-2))) where r = correlation coefficient, n = sample size t-value is used to determine the level of significance of the correlation coefficient
Alternative Hypothesis (Ha) for Correlation Coefficient
Ha: There is a significant relationship between the two variables In other words, the correlation coefficient is not equal to zero
What does the "a" in the regression equation represent?
In the regression equation, "a" represents the intercept, which is the value of the mean of Y when X = 0.
What does the "b" in the regression equation represent?
In the regression equation, "b" represents the slope, which gives the amount of change in the mean of Y (whether positive or negative) for every unit increase in the value of X.
What is regression analysis?
Regression analysis is designed to help determine the probability that inferences are sound by testing the degree to which the dependent variable is affected by the independent variable.
What is the Pearson Product-Moment Correlation Coefficient used for?
The Pearson Product-Moment Correlation Coefficient is used for describing relationships between two variables.
Hypothesis Testing for Correlation Coefficient
Step 1: Set up the null hypothesis (Ho) and alternative hypothesis (Ha) Step 2: Determine the level of significance, degrees of freedom, and critical values (CV) Step 3: Compute the test statistic (t-value) Step 4: Compare the test statistic to the critical values and make a decision whether to reject or fail to reject the null hypothesis Step 5: Make a conclusion based on the decision in Step 4
What is the Correlation Coefficient?
The Correlation Coefficient is the statistic used to report whether two variables are correlated or not and to what extent and in which direction.
What is the Pearson Product-Moment Correlation Coefficient?
The Pearson Product-Moment Correlation Coefficient is the most common correlation measure used for continuous data that are normally distributed.
Testing the Significance
The correlation coefficient tells us the strength and direction of relationship between two variables based on the sample data. However, we are not sure whether the relationship really exists in the population where the sample has been obtained.
What is the formula for computing the PPMC coefficient manually?
The formula for computing the PPMC coefficient manually is: r = [Σ(xy) - (ΣxΣy)/n] / [√(Σx² - (Σx)²/n) × √(Σy² - (Σy)²/n)] where x = value of variable x, y = value of variable y, and n = number of sample points/observations.
What is the range of r values for high correlation?
The range of r values for high correlation is from ±0.80 to +0.99.
What is the range of r values for low correlation?
The range of r values for low correlation is from +0.20 to ±0.39.
What is the range of r values for moderate correlation?
The range of r values for moderate correlation is from ±0.40 to +0.59.
What is the regression equation used for?
The regression equation is an error-free equation used to predict the value of y in perfect correlations. Its formula is: y = a + bx, where a is the intercept (the value of the mean of Y when X = 0) and b is the slope (the amount of change in the mean of Y for every unit increase in the value of X).
What are the three types of correlations based on direction?
The three types of correlations based on direction are: Positive Correlation, Negative Correlation, and Zero/No Correlation.
What is the process for entering data into a Texas calculator for regression analysis?
To enter data into a Texas calculator for regression analysis, first input the data into the calculator's Stat menu and then choose the appropriate variables (i.e., X and Y) and run the regression equation analysis.
What are the different levels of relationship based on the scatter plot?
When the points on the scatter plot do not lie on a straight line, the relationship may be very high, high, moderately high, low, negligible, or zero.
The steps in testing the significance of the correlation coefficient include
computing the correlation coefficient, setting up the null and alternative hypotheses, determining the critical values, computing the test value, making a decision, and drawing a conclusion.
Degrees of Freedom (df) for Correlation Coefficient
df = n - 2, where n = sample size df is used to determine the critical values for the t-test
If the correlation is significant,
then we can conclude that the relationship really exists in the population. There is a need to test the significance of the correlation coefficient to ensure that the high correlation is not just due to chance.