Statistical Methods Statistical Methods Descriptive Statistics Inferential Statistics Estimation...

Statistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

Inference Process

Population

Sample

Sample Statistics

Estimates & Tests

X, Ps

Descriptive Statistics

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

Descriptive Statistics

- Examples of Descriptive Methods -

• HistogramsHistograms

• Bar chartsBar charts

• Average or Arithmetic MeanAverage or Arithmetic Mean

Inferential Statistics

Inferential StatisticsInferential 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

Y Xi i i 0 1

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

Dependent (Response) Variable

Independent (Explanatory) Variable

SlopeY-Intercept

Random Error

i = Random Error

Y

X

Population Linear Regression Model

Observed Value

Observed Value

YX iX 0 1

Y Xi i i 0 1

Sample regression line provides an estimate of the population regression line as well as a predicted value of Y

Sample Linear Regression

Sample Y Intercept

SampleSlopeCoefficient

Residual0 1i iib bY X e

0 1Y b b X 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 estimateestimate of

• b1 provides an estimateestimate of

22

1 1

ˆn n

i i ii i

Y Y e

Comparison of Sample and Population Linear Regression

Y

XObservations

YX iX

i

ii iY X

0 1i iY b b X

ie

0 1i i ib bY X e 1b

0b

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

0

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12000

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Square Feet

An

nu

al S

ales

($0

00)

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

22

,

xxn

yxxynb

xbya

Equation for the Sample Regression Line: Example

0 1ˆ

1636.415 1.487i i

i

Y b b X

X

From Excel Printout:

CoefficientsIntercept 1636.414726X Variable 1 1.486633657

By using the OLS method, we obtained:

Graph of the Sample Regression Line: Example

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0 1000 2000 3000 4000 5000 6000

Square Feet

An

nu

al S

ales

($0

00)

Y i = 1636.415 +1.487X i

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.

ˆ 1636.415 1.487i iY X