1. Math 4030 – 9b Comparing Two Means 1 Dependent and independent samples Comparing two means

2. Choice of test depend on 2 Independent samples design or Matched pairs design (Sec. 8.1) Size of samples Equal variances Normality

3. 3 Independent Samples Matched Pairs Sample One sample of differences (Sec. 8.4) Z test (Sec. 8.2) Large samples? (≥ 30) Y Normality? N Y Equal variance? Y t test with df = n 1 + n 2 -2, using pooled estimator for the common variance (Sec. 8.3) Y t test with estimated degree of freedom (Sec. 8.3) N Nonparametric Tests (Ch.14) N

4. Population 1 (may or may not be normally distributed), with mean  1 (to be estimated and compared) and variance  2 1 (may or may not known). Sample of size n 1 : With sample mean and sample variance: 4 Data format for independent samples: Population 2 (may or may not be normally distributed), with mean  2 (to be estimated and compared) and variance  2 2 (may or may not known). Sample of size n 2 : With sample mean and sample variance:

5. 5 Sampling distribution of : Distribution? CLT still apply?

6. Case 1: both samples are large (n1 ≥ 30, n2 ≥ 30) (Sec. 8.2) 6 or

7. Example 1: 7 It is believed that the resistance of certain electric wire can be reduced by 0.05 ohm by alloying. (Assuming standard deviation of resistance of any wire is 0.035 ohm.) A sample of 32 standard wires and 32 alloyed wires are sampled. Question: Find the probability that average resistance of 32 standard wires is at least 0.03 ohm higher than that of 32 alloyed wires.

8. Confidence interval for : 8 Test statistic for H 0 :

9. Example 2: 9 It is believed that the resistance of certain electric wire can be reduced by alloying. To verify this, a sample of 32 standard wires results the sample mean 0.136 ohm and sample sd 0.034 ohm, and a sample of 32 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.036 ohm Question: Construct a 95% confidence interval for the mean resistance reduction due to alloying.

10. Example 3: 10 It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 32 standard wires results the sample mean 0.136 ohm and sample sd 0.004 ohm, and a sample of 32 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.005 ohm. Question: Can we support the claim at  = 0.05 level?

11. Case 2.1: Small sample(s), normal populations with known equal variance  2 (Sec. 8.3) 11 or

12. Example 4: 12 It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 standard wires results the sample mean 0.136 ohm, and a sample of 15 alloyed wires results the sample mean of 0.083 ohm. (Assume that the resistance has normal distribution with standard deviation 0.0049 ohm for any types of wire.) Question: Can we support the claim at  = 0.05 level?

13. Case 2.2: Small sample(s), normal populations with unknown equal variance (Sec. 8.3) 13

14. Example 5: 14 It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 standard wires results the sample mean 0.136 ohm and sample sd 0.0049, and a sample of 15 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.0052. (Assume that the resistance has normal distribution with the same variance) Question: Can we support the claim at  = 0.05 level?

15. Case 2.3: Small sample(s), normal populations with unequal variance (Sec. 8.3) 15 has t distribution with estimated degree of freedom

16. Only one population, and one sample of size n, but two measurements: 16 Matched Pairs Samples (Sec. 8.4) Since we are interested in the differences, this is really a one sample problem: where

17. 17 Sampling distribution of : Test the hypothesis  D = 0 vs. Confidence interval containing 0.

18. Example 6: 18 It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 wires are tested before the alloying and again after the alloying, we find the mean reduction 0.063 ohm, and the sd of the reductions 0.025. (Assume that the resistance has normal distribution) Question: Can we support the claim at  = 0.05 level?