LESSON 19 CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN

LESSON 19: CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN MEANS Outline • Confidence intervals for the difference between means – Large sample – Small sample equal variance – Small sample unequal variance – Matched pairs 1

DIFFERENCE BETWEEN MEANS • Let, Mean of population A Mean of population B Standard deviation of population A Standard deviation of population B Size of sample drawn from population A Size of sample drawn from population B Mean of the sample drawn from population A Mean of the sample drawn from population B Sample standard deviation, A Sample standard deviation, B 2

DIFFERENCE BETWEEN MEANS • We discuss the computation of confidence interval of • Define the random variable, • Then, 3

LARGE SAMPLE • If each of the two sample sizes is greater than 30, the confidence interval estimate of is computed as follows: 4

DIFFERENCE BETWEEN MEANS Example 1: An engineering society wishes to determine by how much, if at all, the mean outcome of practicing mechanical engineers exceeds that of electrical engineers. Two independent random samples provided the following data: Construct a 90% confidence interval estimate for the difference in population means. , 5

SMALL SAMPLE AND EQUAL VARIANCE • If any of the two sample is less than 30, and both samples have an equal variance, the confidence interval estimate of is computed as follows: 6

SMALL SAMPLE AND EQUAL VARIANCE Example 2: Construct a 99% confidence interval estimate of the difference between the MTBFs of two types of power cells, assuming that the following results have been obtained for independent samples. Assume equal population variances. Cell A Cell B 1 540 days 500 days 2 450 435 3 620 645 4 730 610 5 270 370 6 550 475 7

SMALL SAMPLE AND UNEQUAL VARIANCE • If any of the two sample is less than 30, and the samples have unequal variances, the confidence interval estimate of is computed as follows: 8

SMALL SAMPLE AND UNEQUAL VARIANCE Example 3: Redo Example 2 assuming unequal population variance. 9

MATCHED-PAIR SAMPLES • So far, we discussed procedures that compares means of two samples drawn independently of one another. • However, matched-pair samples require that for every item drawn from one population there be an item drawn from the other population. • For example, suppose that it is required to compare performance of two makes of engines A and B. A random sample of 25 engines are selected from type A engines. Each type A engine is paired with a type B engine, also randomly selected. Each match is based on age and size of engine. Such a sample is a matched-pair sample. 10

MATCHED-PAIR SAMPLES • Letting i denote the i-th pair, with and representing the sample observations for that pair from the respective group, the confidence interval estimate of is computed as follows: 11

MATCHED-PAIR SAMPLES Example 4: Suppose that the observations in Example 2 have been matched according to type of application, environmental condition, and power demands. The pairs are as listed. Construct a 99% confidence interval estimate for the difference in MTBFs between the two types of power cell. 12

READING AND EXERCISES Lesson 19 Reading: Section 10 -5, pp. 319 -330 Exercises: 10 -37, 10 -38, 10 -39, 10 -42 13