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# Chapter 4

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### Chapter 4

1. 1. Chapter 4 Basic Estimation Techniques
2. 2. Simple Linear Regression 4- <ul><li>Simple linear regression model relates dependent variable Y to one independent (or explanatory) variable X </li></ul>• <ul><ul><li>Slope parameter ( b ) gives the change in Y associated with a one-unit change in X , </li></ul></ul>
3. 3. Method of Least Squares 4- <ul><li>The sample regression line is an estimate of the true regression line </li></ul>• •
4. 4. Sample Regression Line (Figure 4.2) 4- A 0 8,000 2,000 10,000 4,000 6,000 10,000 20,000 30,000 40,000 50,000 60,000 70,000 Advertising expenditures (dollars) Sales (dollars) S • • • • • • • e i
5. 5. <ul><li>The distribution of values the estimates might take is centered around the true value of the parameter </li></ul><ul><li>An estimator is unbiased if its average value (or expected value) is equal to the true value of the parameter </li></ul>Unbiased Estimators 4- • •
6. 6. Relative Frequency Distribution* (Figure 4.3) 4- 0 8 2 10 4 6 1 1 3 5 7 9 *Also called a probability density function (pdf)
7. 7. <ul><li>Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e., b  0) </li></ul>Statistical Significance 4- <ul><li>Test for statistical significance using t -tests or p -values </li></ul>•
8. 8. <ul><li>First determine the level of significance </li></ul><ul><ul><li>Probability of finding a parameter estimate to be statistically different from zero when, in fact, it is zero </li></ul></ul><ul><ul><li>Probability of a Type I Error </li></ul></ul><ul><li>1 – level of significance = level of confidence </li></ul>Performing a t -Test 4-
9. 9. Performing a t -Test <ul><li>Use t -table to choose critical t -value with n – k degrees of freedom for the chosen level of significance </li></ul><ul><ul><li>n = number of observations </li></ul></ul><ul><ul><li>k = number of parameters estimated </li></ul></ul>4- •
10. 10. Performing a t -Test <ul><li>If absolute value of t -ratio is greater than the critical t , the parameter estimate is statistically significant </li></ul>4-
11. 11. Using p -Values <ul><li>Treat as statistically significant only those parameter estimates with p -values smaller than the maximum acceptable significance level </li></ul><ul><li>p -value gives exact level of significance </li></ul><ul><ul><li>Also the probability of finding significance when none exists </li></ul></ul>4-
12. 12. Coefficient of Determination <ul><li>R 2 measures the percentage of total variation in the dependent variable that is explained by the regression equation </li></ul><ul><ul><li>Ranges from 0 to 1 </li></ul></ul><ul><ul><li>High R 2 indicates Y and X are highly correlated </li></ul></ul>4-
13. 13. F -Test <ul><li>Used to test for significance of overall regression equation </li></ul><ul><li>Compare F -statistic to critical F -value from F -table </li></ul><ul><ul><li>Two degrees of freedom, n – k & k – 1 </li></ul></ul><ul><ul><li>Level of significance </li></ul></ul><ul><li>If F -statistic exceeds the critical F , the regression equation overall is statistically significant </li></ul>4-
14. 14. Multiple Regression <ul><li>Uses more than one explanatory variable </li></ul><ul><li>Coefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable </li></ul>4-
15. 15. <ul><li>Use when curve fitting scatter plot </li></ul>Quadratic Regression Models 4- • • • is U -shaped or U -shaped
16. 16. Log-Linear Regression Models 4- • • • • •