1. BCOR 1020 Business Statistics Lecture 21 – April 8, 2008

2. Overview Chapter 9 – Hypothesis Testing –Testing a Mean (  ): Population Variance (  ) Unknown –Type I and Type II Errors – Power

3. Chapter 9 – Testing a Mean (  unknown) Hypothesis Tests on  (  unknown): If we wish to test a hypothesis about the mean of a population when  is not assumed to be known, we will follow the same logic as with our previous Z tests… –Specify the level of significance,  (given in problem or assume 5%). –State the null and alternative hypotheses, H 0 and H 1 (based on the problem statement). –Compute the test statistic and determine its distribution under H 0. –State the decision criteria (based on the hypotheses and distribution of the test statistic under H 0 ). –State your decision.

4. Chapter 9 – Testing a Mean (  unknown) Selection of H 0 and H 1 : Remember, the conclusion we wish to test should be stated in the alternative hypothesis. Based on the problem statement, we choose from… (i)H 0 :  >  0 H 1 :  <  0 (ii) H 0 :  <  0 H 1 :  >  0 (iii) H 0 :  =  0 H 1 :   0 where  0 is the null hypothesized value of  (based on the problem statement).

5. Chapter 9 – Testing a Mean (  unknown) Test Statistic: Start with the point estimate of , Recall that for a sample from a normal population with an unknown standard deviation, we can estimate  with the sample standard deviation If H 0 is true and the population from which we are sampling is normal, then the test statistic has the student’s t distribution with = (n – 1) degrees of freedom.

6. Chapter 9 – Testing a Mean (  unknown) Decision Criteria: Just as with previously discussed hypothesis tests, our decision criteria will consist of comparing our test statistic to an appropriate critical point in the student’s t distribution with n – 1 d.f. (the distribution of T* under H 0 ). (i)For the hypothesis test H 0 :  >  0 vs. H 1 :  <  0, we will reject H 0 in favor of H 1 if T* < – t  n . (ii) For the hypothesis test H 0 :   0, we will reject H 0 in favor of H 1 if T* > t  n . (iii) For the hypothesis test H 0 :  =  0 vs. H 1 :   0, we will reject H 0 in favor of H 1 if |T*| > t  /2,n–1.

7. Chapter 9 – Testing a Mean (  unknown) Example: An internet business would like to determine the average income of its customers. In a random sample of 10 customers, the following incomes are reported (in thousands of dollars): 50, 65, 32, 48, 60, 52, 85, 90, 105, 68 Test the appropriate hypothesis to determine whether the mean income of this companies customers exceeds $50,000 at the 10% level of significance.

8. Chapter 9 – Testing a Mean (  unknown) Example (continued): We are hoping to determine whether  > $50,000. We know: The appropriate null and alternative hypotheses are (ii) H 0 :  . We calculate our test statistic (T* since  is unknown and we are using its estimate, S). T* has the student’s t distribution with = (n – 1) = 9 degrees of freedom.

9. t.10,9 = 1.383 Chapter 9 – Testing a Mean (  unknown) Example (continued): Our decision criteria is: (ii) For the hypothesis test H 0 :   0, we will reject H 0 in favor of H 1 if T* > t ,n-1. Since T* = 2.194 > t ,n-1 = 1.383, we reject H 0 in favor of H 1 :  > $50,000. Based on the observed data, there is statistically significant evidence that the mean customer income exceeds $50,000.

10. T* = 2.194 Chapter 9 – Testing a Mean (  unknown) Example (continued): Using Megastat, we can calculate an exact p-value for this test based on the data: p-value = P(t ,9 > 2.194) = 0.0279 Interpretation: If we were to reject H 0 based on the observed data, there is less than a 3% chance we would be making a type I error. Since this is smaller than  = 10%, we will reject H 0.

11. Problem: Testing A Hypothesis on a Mean (  Unknown) In an effort to get a loan, a clothing retailer has made the claim that the average daily sales at her store exceeds $7500. Suppose we have no information about the true value of the standard deviation, . In a randomly selected sample of 24 days sales data, the average daily sales were found to be = $7900 with a standard deviation of S = $1500. At the 10% level of significance, conduct the appropriate hypothesis test to determine whether the data supports the retailer’s claim. Include the following: –State the level of significance, . –State the null and alternative hypotheses, H 0 and H 1. –Compute the test statistic –State the decision criteria –State your decision (Overhead)

12. Problem: Testing A Hypothesis on a Mean (  Unknown) Work:  = H 0 : H 1 : Test Statistic (and Distribution under H 0 )… Decision Criteria… Decision…

13. Clickers What are the appropriate null and alternative hypotheses? (C) H 0 :  >  H 1 :  <  (B) H 0 :  <  H 1 :  >  (A) H 0 :  =  H 1 :  

14. Clickers What is the calculated value of your test statistic? (A) T* = 0.267 (B) T* = 1.306 (C) T* = 1.645 (D) T* = 1.963

15. Clickers What is your decision criteria? (A) Reject H 0 if T* > 1.319 (B) Reject H 0 if T* > 1.645 (C) Reject H 0 if T* > 1.714 (D) Reject H 0 if T* > 1.960

16. Clickers What is your decision? (A) Reject H 0 in favor of H 1 (B) Fail to reject (Accept) H 0 in favor of H 1 (C) There is not enough information

17. Problem: Testing A Hypothesis on a Mean (  Unknown) Conclusion: Since our test statistic T* = 1.306 is not greater than t  = 1.319, we will fail to reject H 0 in favor of H 1 :  > $7500. Based on the data in this sample, there is not statistically significant evidence that the average daily sales at her store exceeds $7500. What is the p-value in a T Test? –We can bound it using the t-table (n – 1 d.f. row). –For this problem t.15,23 = 1.060 < T* < t.10,23 = 1.319 –So, 0.10 < p-value < 0.15. –If we reject H 0 based on this sample, the probability of a type I error is between 10% and 15%.

18. Chapter 9 – Type I & Type II Errors Type II Error: , the probability of a type II error, is the probability that the test statistic falls in the acceptance region even though H 0 is false.  = P(fail to reject H 0 | H 0 is false)  cannot be chosen in advance because it depends on  and the sample size. A small  is desirable, other things being equal.

19. Chapter 9 – Type I & Type II Errors Power of a Test: The power of a test is the probability that a false hypothesis will be rejected. Power = 1 –  A low  risk means high power. Larger samples lead to increased power. Power = P(reject H 0 | H 0 is false) = 1 – 

20. Chapter 9 – Type I & Type II Errors Relationship Between  and  : Both a small  and a small  are desirable. For a given type of test and fixed sample size, there is a trade-off between  and . The larger critical value needed to reduce  risk makes it harder to reject H 0, thereby increasing  risk. Both  and  can be reduced simultaneously only by increasing the sample size.

21. Chapter 9 – Type I & Type II Errors Power depends on how far the true value of the parameter is from the null hypothesis value. The further away the true population value is from the assumed value, the easier it is for your hypothesis test to detect and the more power it has. Remember that Power Curves for a Mean:  = P(fail to reject H 0 | H 0 is false) Power = P(reject H 0 | H 0 is false) = 1 – 

22. Chapter 9 – Type I & Type II Errors We want power to be as close to 1 as possible. The values of  and power will vary, depending on - the difference between the true mean  and the hypothesized mean  0, - the standard deviation, - the sample size n and - the level of significance  Power = f (  –  0, , n,  ) Power Curves for a Mean:

23. Chapter 9 – Type I & Type II Errors We can get more power by increasing , but we would then increase the probability of a type I error. A better way to increase power is to increase the sample size n. Power Curves for a Mean:

24. Chapter 9 – Type I & Type II Errors Calculating Power: We will start by calculating the Type II error of our hypothesis test. In order to do this, we need to know (or assume) the true value of the parameter if, in fact, the Alternative hypothesis is true. Consider the test where H 1 :  >  0 … H 0 :  <  0 H 1 :  >  0 ; specifically,  =  a =  0 +   is the difference between the null hypothesized value and the true value of the mean.

25. Chapter 9 – Type I & Type II Errors Calculating Power: Details of the calculation by hand…

26. Chapter 9 – Type I & Type II Errors Calculating Power: For the “>” alternative hypothesis… We can follow a similar process to find formulas for Type II error for the other two alternatives as well.

27. Chapter 9 – Type I & Type II Errors Calculating Power: For either the “ ” alternative hypotheses… For the “ ” alternative hypothesis… Where  = |  0 –  a | is the magnitude of the difference between the null hypothesized value and the true value. Power = 1 - 

28. Chapter 9 – Type I & Type II Errors We can use these equations to determine sample size requirements for specified values of  and . For either the “ ” alternative hypotheses… For the “ ” alternative hypothesis… Always round up.

29. Chapter 9 – Type I & Type II Errors Example… In an effort to get a loan, a clothing retailer has made the claim that the average daily sales at her store exceeds $7500. Historical data suggests that the purchase amount is normally distributed with a standard deviation of  = $1500. In a randomly selected sample of 24 days sales data, the average daily sales were found to be $7900. A hypothesis test is conducted to determine whether the data supports the retailer’s claim at the 10% level. Find the power of this test if the true average daily sales are $8000.

30. Chapter 9 – Type I & Type II Errors Example (continued)…  = 0.10; Z  = Z.10 = 1.282  = |  0 –  a | = |$7500 - $8000| = $500  = $1500 n = 24 Power = 1 –  = 0.6368 There is an approximately 64% chance that this test will correctly identify that the mean exceeds $7500 by $500.

31. Chapter 9 – Type I & Type II Errors Example (continued)… If we would like to increase the power to 75%, how large should our sample be (all other things being equal)?  = 0.10; Z  = Z.10 = 1.282  = 0.25; Z  = Z.25 = 0.675  = |  0 –  a | = |$7500 - $8000| = $500  = $1500

32. Clickers Suppose you are testing the hypothesis H 0 :  > 80 vs. H 1 :  < 80. Which of the following probability statements correspond to the Power of the test? Reject H 0 when H 0 False – Correct Decision! POWER Reject H 0 when H 0 True – Type I Error! Fail to reject H 0 when H 0 False – Type II Error! Fail to reject H 0 when H 0 True – Correct Decision! 90 seconds…