Statistical Methods Statistical Methods Descriptive Inferential Statistics Statistics Hypothesis Estimation Testing

Inference Process Estimates & Tests Population Sample Statistics X, Ps Sample

Descriptive Statistics Descriptive Statistics consists of the tools and techniques designed to describe data, such as charts, graphs, and numerical measures.

Descriptive Statistics - Examples of Descriptive Methods - • Histograms • Bar charts • Average or Arithmetic Mean

Inferential Statistics Inferential Statistics consists of techniques that allow a decision-maker to reach a conclusion about characteristics of a larger data set based upon a subset of those data

Statistical Inferences Simple Linear Regression

Population Linear Regression Model Relationship between variables is described by a linear function The change of the independent variable causes the change in the dependent variable Random Error Slope Y-Intercept  Y     X  i 1 i i Dependent (Response) Variable Independent (Explanatory) Variable

Population Linear Regression Model Y Y     X   Observed Value i 1 i i  = Random Error i      X YX 1 i X Observed Value

Sample Linear Regression Sample regression line provides an estimate of the population regression line as well as a predicted value of Y Sample Y Intercept Sample Slope Coefficient Residual Sample Regression Line (Fitted Regression Line)

Sample Linear Regression Using Ordinary Least Squares (OLS), we can find the values of b0 and b1 that minimize the sum of the squared residuals: b0 provides an estimate of b1 provides an estimate of

Comparison of Sample and Population Linear Regression Y X Observations

Simple Linear Regression: Example You want to examine the linear dependency of the annual sales of produce stores on their size in square footage. Sample data for seven stores were obtained. Find the equation of the straight line that fits the data best. X Y Annual Store Square Sales Feet ($1000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760

Scatter Diagram: Example Excel Output

Least Squares Estimates Using calculus (partial derivatives), we get Note b is related to the correlation coefficient r (same numerator)- if x and y are positively correlated then the slope is positive

Equation for the Sample Regression Line: Example By using the OLS method, we obtained: From Excel Printout:

Graph of the Sample Regression Line: Example Yi = 1636.415 +1.487Xi 

Interpretation of Results: Example The slope of 1.487 means that for each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units. The model estimates that for each increase of one square foot in the size of the store, the expected annual sales are predicted to increase by $1.487.