Correlation and Linear Regression Chapter 13

Learning Objectives LO13-1 Explain the purpose of correlation analysis. LO13-2 Calculate a correlation coefficient to test and interpret the relationship between two variables. LO13-3 Apply regression analysis to estimate the linear relationship between two variables. (excluded) LO13-4 Evaluate the significance of the slope of the regression equation. (excluded) LO13-5 Evaluate a regression equation’s ability to predict using the standard estimate of the error and the coefficient of determination. (excluded) LO13-6 Calculate and interpret confidence and prediction intervals. (excluded) LO13-7 Use a log function to transform a nonlinear relationship. (excluded)

LO13-1 Explain the purpose of correlation analysis. Correlation Analysis – Measuring the Relationship Between Two Variables Analyzing relationships between two quantitative variables. The basic hypothesis of correlation analysis: Does the data indicate that there is a relationship between two quantitative variables? For the Applewood Auto sales data, the data is displayed in a scatter graph. Are profit per vehicle and age correlated?

Correlation Analysis – Measuring LO13-1 Correlation Analysis – Measuring the Relationship Between Two Variables The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. The sample correlation coefficient is identified by the lowercase letter r. It shows the direction and strength of the linear relationship between two interval- or ratio-scale variables. It ranges from -1 up to and including +1. A value near 0 indicates there is little linear relationship between the variables. A value near +1 indicates a direct or positive linear relationship between the variables. A value near -1 indicates an inverse or negative linear relationship between the variables.

LO13-1 Correlation Analysis – Measuring the Relationship Between Two Variables .

Computing the Correlation Coefficient: LO13-2 Calculate a correlation coefficient to test and interpret the relationship between two variables. Correlation Analysis – Measuring the Relationship Between Two Variables Computing the Correlation Coefficient:

Computing the Correlation Coefficient – Example LO13-2 Computing the Correlation Coefficient – Example The sales manager of Copier Sales of America has a large sales force throughout the United States and Canada and wants to determine whether there is a relationship between the number of sales calls made in a month and the number of copiers sold that month. The manager selects a random sample of 15 representatives and determines the number of sales calls each representative made last month and the number of copiers sold. Determine if the number of sales calls and copiers sold are correlated.

Computing the Correlation Coefficient – Example Inspect a scatter diagram

Computing the Correlation Coefficient - Example

Testing the significance of the Correlation Coefficient LO13-2 Testing the significance of the Correlation Coefficient Step 1: State the null and alternate hypotheses. H0:  = 0 (the correlation in the population is 0) H1:  ≠ 0 (the correlation in the population is not 0) Step 2: Select a level of significance. We select a .05 level of significance. Step 3: Identify the test statistic. To test a hypothesis about a correlation we use the t-statistic. For this analysis, there will be n-2 degrees of freedom.

Testing the significance of the Correlation Coefficient LO13-2 Testing the significance of the Correlation Coefficient Step 4: Formulate a decision rule. Reject H0 if: t > t/2,n-2 or t < -t/2,n-2 t > t0.025,13 or t < -t0.025,13 t > 2.160 or t < -2.160

Testing the significance of the Correlation Coefficient LO13-2 Testing the significance of the Correlation Coefficient Step 5 (continued): Take a sample, calculate the statistics, arrive at a decision. The t-test statistic, 6.216, is greater than 2.160. Therefore, reject the null hypothesis that the correlation coefficient is zero. Step 6: Interpret the result. The data indicate that there is a significant correlation between the number of sales calls and copiers sold. We can also observe that the correlation coefficient is .865, which indicates a strong, positive relationship. In other words, more sales calls are strongly related to more copier sales. Please note that this statistical analysis does not provide any evidence of a causal relationship. Another type of study is needed to test that hypothesis.