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**Forecasting Using the Simple Linear Regression Model and Correlation**

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What is a forecast? Using a statistical method on past data to predict the future. Using experience, judgment and surveys to predict the future.

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**Why forecast? to enhance planning. to force thinking about the future.**

to fit corporate strategy to future conditions. to coordinate departments to the same future. to reduce corporate costs.

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Kinds of Forecasts Causal forecasts are when changes in a variable (Y) you wish to predict are caused by changes in other variables (X's). Time series forecasts are when changes in a variable (Y) are predicted based on prior values of itself (Y). Regression can provide both kinds of forecasts.

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**Types of Relationships**

Positive Linear Relationship Negative Linear Relationship

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**Types of Relationships**

(continued) Relationship NOT Linear No Relationship

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Relationships If the relationship is not linear, the forecaster often has to use math transformations to make the relationship linear.

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Correlation Analysis Correlation measures the strength of the linear relationship between variables. It can be used to find the best predictor variables. It does not assure that there is a causal relationship between the variables.

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**The Correlation Coefficient**

Ranges between -1 and 1. The Closer to -1, The Stronger Is The Negative Linear Relationship. The Closer to 1, The Stronger Is The Positive Linear Relationship. The Closer to 0, The Weaker Is Any Linear Relationship.

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**Graphs of Various Correlation (r) Values**

Y Y Y X X X r = -1 r = -.6 r = 0 Y Y X X r = .6 r = 1

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The Scatter Diagram Plot of all (Xi , Yi) pairs

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The Scatter Diagram Is used to visualize the relationship and to assess its linearity. The scatter diagram can also be used to identify outliers.

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Regression Analysis Regression Analysis can be used to model causality and make predictions. Terminology: The variable to be predicted is called the dependent or response variable. The variables used in the prediction model are called independent, explanatory or predictor variables.

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**Simple Linear Regression Model**

The relationship between variables is described by a linear function. A change of one variable causes the other variable to change.

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**Population Linear Regression**

Population Regression Line Is A Straight Line that Describes The Dependence of One Variable on The Other Population Slope Coefficient Random Error Population Y intercept Dependent (Response) Variable Population Regression Line Independent (Explanatory) Variable

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**How is the best line found?**

Y Observed Value = Random Error X Observed Value

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**Sample Linear Regression**

Sample Regression Line Provides an Estimate of The Population Regression Line Sample Slope Coefficient Sample Y Intercept Residual Sample Regression Line provides an estimate of provides an estimate of

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**Simple Linear Regression: An Example**

Annual Store Square Sales Feet ($1000) , ,681 , ,395 , ,653 , ,543 , ,318 , ,563 , ,760 You wish to examine the relationship between the square footage of produce stores and their annual sales. Sample data for 7 stores were obtained. Find the equation of the straight line that fits the data best

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The Scatter Diagram Excel Output

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**The Equation for the Regression Line**

From Excel Printout:

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**Graph of the Regression Line**

Yi = Xi

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**Interpreting the Results**

Yi = Xi The slope of means that each increase of one unit in X, we predict the average of Y to increase by an estimated units. The model estimates that for each increase of 1 square foot in the size of the store, the expected annual sales are predicted to increase by $1487.

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**The Coefficient of Determination**

SSR regression sum of squares r2 = = SST total sum of squares The Coefficient of Determination (r2 ) measures the proportion of variation in Y explained by the independent variable X.

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**Coefficients of Determination (R2) and Correlation (R)**

Y ^ Y = b + b X i 1 i X

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**Coefficients of Determination (R2) and Correlation (R)**

(continued) r2 = .81, r = +0.9 Y ^ Y = b + b X i 1 i X

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**Coefficients of Determination (R2) and Correlation (R)**

(continued) r2 = 0, r = 0 Y ^ Y = b + b X i 1 i X

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**Coefficients of Determination (R2) and Correlation (R)**

(continued) r2 = 1, r = -1 Y ^ Y = b + b X i 1 i X

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**Correlation: The Symbols**

Population correlation coefficient (‘rho’) measures the strength between two variables. Sample correlation coefficient r estimates based on a set of sample observations.

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**Example: Produce Stores**

From Excel Printout

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**Inferences About the Slope**

t Test for a Population Slope Is There A Linear Relationship between X and Y ? Null and Alternative Hypotheses H0: 1 = 0 (No Linear Relationship) H1: 1 0 (Linear Relationship) Test Statistic: Where and df = n - 2

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**Example: Produce Stores**

Data for 7 Stores: Estimated Regression Equation: Annual Store Square Sales Feet ($000) , ,681 , ,395 , ,653 , ,543 , ,318 , ,563 , ,760 Yi = Xi The slope of this model is Is Square Footage of the store affecting its Annual Sales?

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**Inferences About the Slope: t Test Example**

Test Statistic: Decision: Conclusion: H0: 1 = 0 H1: 1 0 .05 df = 5 Critical value(s): From Excel Printout Reject Reject Reject H0 .025 .025 There is evidence of a linear relationship. t 2.5706

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**Inferences About the Slope Using A Confidence Interval**

Confidence Interval Estimate of the Slope b1 tn-2 Excel Printout for Produce Stores At 95% level of Confidence The confidence Interval for the slope is (1.062, 1.911). Does not include 0. Conclusion: There is a significant linear relationship between annual sales and the size of the store.

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Residual Analysis Is used to evaluate validity of assumptions. Residual analysis uses numerical measures and plots to assure the validity of the assumptions.

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**Linear Regression Assumptions**

1. X is linearly related to Y. 2. The variance is constant for each value of Y (Homoscedasticity). 3. The Residual Error is Normally Distributed. 4. If the data is over time, then the errors must be independent.

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**Residual Analysis for Linearity**

X X e e X X Not Linear Linear

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**Residual Analysis for Homoscedasticity**

X X e e X X Homoscedasticity Heteroscedasticity

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**Residual Analysis for Independence: The Durbin-Watson Statistic**

It is used when data is collected over time. It detects autocorrelation; that is, the residuals in one time period are related to residuals in another time period. It measures violation of independence assumption. Calculate D and compare it to the value in Table E.8.

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**Preparing Confidence Intervals for Forecasts**

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**Interval Estimates for Different Values of X**

Confidence Interval for the mean of Y Confidence Interval for a individual Yi Y Yi = b0 + b1Xi _ X X A Given X

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**Estimation of Predicted Values**

Confidence Interval Estimate for YX The Mean of Y given a particular Xi Size of interval vary according to distance away from mean, X. Standard error of the estimate t value from table with df=n-2

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**Estimation of Predicted Values**

Confidence Interval Estimate for Individual Response Yi at a Particular Xi Addition of 1 increases width of interval from that for the mean of Y

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**Example: Produce Stores**

Data for 7 Stores: Annual Store Square Sales Feet ($000) , ,681 , ,395 , ,653 , ,543 , ,318 , ,563 , ,760 Predict the annual sales for a store with 2000 square feet. Regression Model Obtained: Yi = Xi

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**Estimation of Predicted Values: Example**

Confidence Interval Estimate for YX Find the 95% confidence interval for the average annual sales for stores of 2,000 square feet Predicted Sales Yi = Xi = ($000) tn-2 = t5 = X = SYX = = Confidence interval for mean Y

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**Estimation of Predicted Values: Example**

Confidence Interval Estimate for Individual Y Find the 95% confidence interval for annual sales of one particular store of 2,000 square feet Predicted Sales Yi = Xi = ($000) tn-2 = t5 = X = SYX = = Confidence interval for individual Y

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