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BA 275 Quantitative Business Methods

BA 275 Quantitative Business Methods. Agenda. Statistical Inference: Hypothesis Testing Type I and II Errors Power of a Test Hypothesis Testing Using Statgraphics. Type I, II Errors , and Power. a = P( Type I Error ) = P( reject H 0 given that H 0 is true)

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BA 275 Quantitative Business Methods

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  1. BA 275 Quantitative Business Methods Agenda • Statistical Inference: Hypothesis Testing • Type I and II Errors • Power of a Test • Hypothesis Testing Using Statgraphics

  2. Type I, II Errors, and Power a = P( Type I Error ) = P( reject H0 given that H0 is true) b = P( Type II Error ) = P( fail to reject H0 given that H0 is false) Power = 1 – b = P( reject H0 given that H0 is false ) Refer to textbook p. 416

  3. Example 1 • You want to see if a redesign of the cover of a mail-order catalog will increase sales. Assume that the mean sales from the new catalog will be approximately normal with s = $50 and that the mean for the original catalog will be m = $25. Given a sample of size n = 900, you wish to test H0: m = 25 vs. Ha: m > 25. Rejection region is: reject H0 if the sample mean > 26. • Q1: Find a = P( Type I error ). • Q2: Find b = P( Type II error ) given m = $28. • Q3: Find b = P( Type II error ) given m = $30.

  4. Example 2 • A survey of 100 retailers was conducted to see if the mean after-tax profit exceeded $70,000. Assume that the population standard deviation is $20,000. • Q1. Let rejection region be: Rejection H0 if the sample mean exceeds $73,290. Find a. • Q2. Calculate the power of the test given that the true m = 75,500? • Q3. Suppose the sample mean is $75,000. Find the p-value. • Q4. Estimate the true mean after-tax profit using a 95% confidence interval.

  5. Hypothesis Testing Using SG • Steps are similar to what you did for Confidence Interval Estimation. • Use the right mouse button for more options. For example, to change the significance level (the value of a) and/or to select the type of tests (lower, upper, or two-tailed test.) • Refer to page 3 of the SG instruction for interval estimation (filename: SG Instruction Estimation.doc downloadable from the class website) on how to select a subset of sample for analysis.

  6. Central Limit Theorem (CLT) • The CLT applied to Means With a large n, approximate s with s. The CLT still holds. What if s is unknown? What if s is unknown and n is small? Need to modify the CLT.

  7. Central Limit Theorem (CLT) • s is unknown but n is large

  8. Central Limit Theorem (CLT) • s is unknown and n is small

  9. T distribution with degrees of freedom 5 vs. Normal(0, 1)

  10. Example 3 • A random sample of 10 one-bedroom apartments (Ouch, a small sample) from your local newspaper gives a sample mean of $541.5 and sample standard deviation of $69.16. Assume a = 5%. • Q1. Does the sample give good reason to believe that the mean rent of all advertised apartments is greater than $500 per month? (Need H0 and Ha, rejection region and conclusion.) • Q2. Find the p-value. • Q3. Construct a 95% confidence interval for the mean rent of all advertised apartments. • Q4. What assumption is necessary to answer Q1-Q3.

  11. T Table (Table D)

  12. Answer Key to Examples UsedExample 1

  13. Answer Key to Examples UsedExample 2

  14. Answer Key to Examples UsedExample 3

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