Business Statistics: Chapter 13; Correlation and Linear Regression

Correlation Analysis

    Correlation Analysis: group of techniques used to measure the relationship between two variables. First step is to plot data in a scatter diagram.

    Scatter Diagram: used to portray the relationship between two variables.

  • Independent Variable: A variable that provides the basis for estimation. The x-values.

  • Dependent Variable: The variable that is being predicted or estimated.. The y-value.

Correlation Coefficient

    Correlation Coefficient: describes strength of relationship between two sets of interval- or ratio-scaled variables.

        r = (each x minus the mean of x) + (each y minus the mean of y) / (n-1) * standard deviation of x *               standard deviation of y

    Characteristic of the Correlation Coefficient:

  • Sample correlation coefficient is identified by r

  • Shows direction and strength of linear relationship between two interval- or ratio-scale variables

  • Ranges from -1 to +1

  • Value near 0 shows little relationship between variables

  • Value near +1 shows direct or positive relationship between variables

  • Value near -1 shows inverse or negative relationship between variables

    Excel Functions:

  • Use data analysis, descriptive statistics, select data for data like mean, sample standard deviation, etc.

  • To compute correlation coefficient, r, use data analysis, correlation, then select data set for r.

    Testing Significance of Correlation Coefficient, r:

  • t-test for correlation coefficient:

            t = r * square root of n-2 / square root of 1-r² with n-2 degrees of freedom

  • Decision rule, fine critical value, use t-table with degrees of freedom and significance level. If two-tailed, remember df=a/2.

Regression Analysis

    Regression Equation: expressed relationship between two variables. Use to estimate one variable based on another, usually y based on x, or independent based on dependent variable.

        ŷ= estimated y-value for selected x-value= a +bx

            Where a is the y-intercept, b is the slope, and x is the independent variable selected

                b = slope = r * (sample standard deviation of y / sample standard deviation of x)

                a = y-intercept = ȳ - bx̄

                    ȳ is mean of y, x̄ is mean of x


    Lead Squares Principle: use data to draw a line to minimize sum of squares of vertical distance between actual y-values and predicted y-values.

Testing Significance of the Slope

    Test for Slope: test hypothesis that slope of line is 0. If rejected null hypothesis, then there is a relationship between the variables. If accept null, then there is no relationships.

  • t = b - 0 / standard error of slope estimate

  • To find decision rule, use t-table with significance levels and degrees of freedom (n - 2)

Evaluating Regression Equation’s Ability to Predict

    Standard Error of Estimate: measures of scatter of observed values around line of regression for a given value of x.

        = square root of all y - (mean of y)² added / n - 2 = square root of ( SSE / n - 2 = square root of MSR

    Coefficient of Determination: proportion of total variation in the dependent variable y that is explained by the variation in independent variable x.

        r² = SSR / SS Total = 1 - (SSE / SS Total)

    Relationships Between Correlation Coefficient, Coefficient of Determination, and Standard Error of Estimate:

  • Standard error of estimate measures how close the actual values are to the regression line. When it is small, the two variables are closely related.

  • Correlation coefficient measures the strength of the linear association between two variables. When points on the scatter diagram are close to the line, the correlation coefficient tends to be large.

  • Correlation coefficient and the standard error of estimate are inversely related

  • The coefficient of determination is the correlation coefficient squared

Interval Estimates of Prediction

    Regression Assumptions:

  • Follow the normal distribution

  • Has a mean on the regression line

  • Has the same standard error of estimate

  • Is independent of the others

    Inference about Linear Regression:

  • ŷ + and - standard error of estimate will include 68% of observations

  • ŷ + and - 2 * standard error of estimate will include 68% of observations will include 95% of observations

  • ŷ + and - 3 * standard error of estimate will include 68% of observations will include virtually all observations