Selecting Two Vouchers Lecture 5 Section 1.5 Fri, Jan 23, 2004

Sample Size n = 2 Consider again the original Bag A and Bag B. Now let the sample size be 2. What are the possible samples? How likely are they? We will sample without replacement. Therefore, the two vouchers selected must be different.

Sample Size n = 2 Consider Bag A (see p. 40). There are 20 vouchers in the bag. So there are 20 ways to choose the first voucher. Once the first voucher is chosen, there are 19 vouchers in the bag. So there are 19 ways to choose the second voucher.

Sample Size n = 2 Thus, there are 20 x 19 = 380 ways to choose two vouchers. However! This counts each pair of vouchers twice. For example: ($10, $30) and ($30, $10). Thus, there are 380  2 = 190 distinct ways.

The Sampling Distribution See pages 42, 43, and 44 for charts and diagrams showing the distributions of sample averages for Bag A and Bag B. The diagrams show averages, not totals. Don’t worry. We will not try to do this for n = 3.

A Decision Rule Let’s use Decision Rule #2 on p. 46: What is ? Reject H0 if the selected average is  $50. What is ?  = 3/190 = 0.0158. What is ?  = 79/190 = 0.416.

Let’s Do It! Let’s do it! 1.12 – Finding a 5% Level Decision Rule. What is 5% of 190? What value (of the average) cuts off that many samples in the right tail?

The Effect of Increasing the Sample Size Compare the values of  when  = 0.05, for the two sample sizes. When n = 1,  = 12/20 = 0.60. When n = 2,  = 53/190 = 0.279.

The Effect of Increasing the Sample Size Compare the values of  when  = 0.10. When n = 1,  = 6/20 = 0.30. When n = 2,  = 33/190 = 0.174.

The Effect of Increasing the Sample Size By increasing the sample size, we can reduce both  and  simultaneously. For large samples, there is very little risk of making either a Type I or a Type II error. Question: How large is large enough?

Assignment Page 54: Exercises 54 – 57.