Chapter 10 Comparing Two Means and Two Proportions

Chapter 10Comparing Two Means and Two ProportionsCopyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinStatistical Inferences Based on Two Samples10.1 Comparing Two Population Means by Using Independent Samples10.2 Paired Difference Experiments10.3 Comparing Two Population Proportions by Using Large, Independent Samples10-*10.1 Comparing Two Population Means by Using Independent SamplesSuppose a random sample has been taken from each of two different populationsSuppose that the populations are independent of each otherThen the random samples are independent of each otherThen the sampling distribution of the difference in sample means is normally distributedLO10-1: Compare two population means when the samples are independent.10-*Sampling Distribution of the Difference of Two Sample Means #1Suppose population 1 has mean µ1 and variance σ12From population 1, a random sample of size n1 is selected which has mean 1 and variance s12Suppose population 2 has mean µ2 and variance σ22From population 2, a random sample of size n2 is selected which has mean 2 and variance s22Then the sample distribution of the difference of two sample meansLO10-110-*Sampling Distribution of the Difference of Two Sample Means #2Is normal, if each of the sampled populations is normalApproximately normal if the sample sizes n1 and n2 are largeHas mean µx1–x2 = µ1 – µ2Has standard deviationLO10-110-*t-Based Confidence Interval for the Difference in Means: Equal VariancesLO10-110-*t-Based Test About the Difference in Means: Equal VarianceLO10-110-*10.3 Paired Difference ExperimentsBefore, drew random samples from two different populationsNow, have two different processes (or methods)Draw one random sample of units and use those units to obtain the results of each processLO10-2: Recognize when data come from independent samples and when they are paired.10-*Paired Difference Experiments ContinuedFor instance, use the same individuals for the results from one process vs. the results from the other processE.g., use the same individuals to compare “before” and “after” treatmentsUsing the same individuals, eliminates any differences in the individuals themselves and just comparing the results from the two processesLO10-210-*t-Based Confidence Interval for Paired Differences in MeansIf the sampled population of differences is normally distributed with mean dA (1- )100% confidence interval forµd = µ1 - µ2 iswhere for a sample of size n, t/2 is based on n – 1 degrees of freedomLO10-3: Compare twopopulation means when the data are paired.10-*Test Statistic for Paired DifferencesLO10-310-*10.3 Comparing Two Population Proportions by Using Large, Independent SamplesSelect a random sample of size n1 from a population, and let p̂1 denote the proportion of units in this sample that fall into the category of interestSelect a random sample of size n2 from another population, and let p̂2 denote the proportion of units in this sample that fall into the same category of interestSuppose that n1 and n2 are large enoughn1·p1≥5, n1·(1 - p1)≥5, n2·p2≥5, and n2·(1 – p2)≥5LO10-4: Compare twopopulation proportions using large independentsamples.10-*Comparing Two Population Proportions ContinuedThen the population of all possible values of p̂1 - p̂2Has approximately a normal distribution if each of the sample sizes n1 and n2 is largeHere, n1 and n2 are large enough so n1·p1 ≥ 5,n1· (1 - p1) ≥ 5, n2·p2 ≥ 5, and n2·(1 – p2) ≥ 5Has mean µp̂1 - p̂2 = p1 – p2Has standard deviationLO10-410-*