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

2. 2 Type I, II Errors, and PowerErrorsPower = P( Type I Error ) = P( reject H 0 given that H 0 is true) = P( Type II Error ) = P( fail to reject H 0 given that H 0 is false) Power = 1 – = P( reject H 0 given that H 0 is false ) Refer to textbook p. 416

3. 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 = $50 and that the mean for the original catalog will be = $25. Given a sample of size n = 900, you wish to test H 0 : = 25 vs. H a : > 25. Rejection region is: reject H 0 if the sample mean > 26. Q1: Find = P( Type I error ). Q2: Find = P( Type II error ) given = $28. Q3: Find = P( Type II error ) given = $30.

4. 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. Q2. Calculate the power of the test given that the true = 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. 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. 6 Central Limit Theorem (CLT) The CLT applied to Means What if is unknown? With a large n, approximate with s. The CLT still holds. What if is unknown and n is small? Need to modify the CLT.

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

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

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

10. 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 = 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. 11 T Table (Table D)

12. 12 Answer Key to Examples Used Example 1

13. 13 Answer Key to Examples Used Example 2

14. 14 Answer Key to Examples Used Example 3