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Simple Linear Regression: Ŷ b 0 + b 1 X 1

Simple Linear Regression: Ŷ b 0 + b 1 X 1. 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.

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Simple Linear Regression: Ŷ b 0 + b 1 X 1

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  1. Simple Linear Regression: Ŷ b0 + b1 X1 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. 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

  2. Which is the dependent Y variable? • The Store Number • The Square Footage of the Store • The Annual Sales of the Store

  3. Which is the independent X variable? • The Store Number • The Square Footage of the Store • The Annual Sales of the Store

  4. Scatter Diagram: Example Excel Output

  5. Equation for the Sample Regression Line: Example From Excel Printout:

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

  7. Interpretation of Results: 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 $1487. If a new 2,000 square foot produce store is built, the model predicts that the expected annual sales at the store will be: 1636 + 1.487*2000 = $4,610 (in 1000s)

  8. The Coefficient of Determination • r2 = 94.2% • Measures the proportion of variation in Y (e.g. Annual Sales) that is explained by the linear regression model with the independent variable X (e.g. square feet) • Describes the explanatory power of the simple linear regression model; it does not imply that X causes the changes in Y.

  9. Significant Coefficients • If a linear relationship between the dependent and independent variables does not exist, the true value of the slope should be 0. To test to see if this is true, look at the 95% confidence interval for an independent variable’s coefficient:

  10. Multiple Linear Regression Model with n independent variables • Equation: • Adjusted R-square: • Describes the explanatory power of the multiple regression, after compensating for sample size and the number of independent variables in the model Ŷi b0 + b1 X1 + b2 X2 +…..+ bnXn

  11. Restaurant Sales Exercise (ChiliDogRegress.xlsx) • The ChiliDog Hut fast food chain wants to identify a good location for a new restaurant • They have identified three possible independent variables that could have a relationship with the annual sales (in $1,000s) of a restaurant • # of other fast food stores in 1 mile radius • # of schools and businesses in 1 mile radius • $ spent on advertising per year • Help ChiliDog identify the regression model that forecasts annual sales the best

  12. Simple Linear Regression on # of Other Restaurants in 1 mile radius

  13. What Conclusions Can You Make From the Simple Linear Regression? • More competing restaurants in the 1 mile radius hurt ChiliDog Hut’s sales • The # of competing restaurants in the 1 mile radius have little impact on ChiliDog Hut’s sales • There is a positive correlation between the # of competing restaurants in the 1 mile radius and ChiliDog Hut’s sales • Increasing the # of competing restaurants in the 1 mile radius will increase ChiliDog Hut’s sales

  14. Multiple Regression on # of Other Restaurants & # of Schools and Businesses in 1 mile radius

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