Copyright © 2014 by McGraw-Hill Higher Education. All rights reserved. Essentials of Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly Copyright © 2014 by McGraw-Hill Higher Education. All rights reserved.

Chapter 10 Learning Objectives LO 10.1 Make inferences about the difference between two population means based on independent sampling. LO 10.2 Make inferences about the mean difference based on matched-pairs sampling. LO 10.3 Discuss features of the F distribution. LO 10.4 Make inferences about the difference between three or more population means using an analysis of variance (ANOVA) test. Statistical Inference Concerning Two Populations

10.1 Inference Concerning the Difference Between Two Means LO 10.1 Make inferences about the difference between two population means based on independent sampling. Independent Random Samples Two (or more) random samples are considered independent if the process that generates one sample is completely separate from the process that generates the other sample. The samples are clearly delineated. m1 is the mean of the first population. m2 is the mean of the second population. Statistical Inference Concerning Two Populations

10.1 Inference Concerning the Difference Between Two Means LO 10.1 Confidence Interval for m1  m2 is a point estimator of m1  m2. The values of the sample means and are computed from two independent random samples with n1 and n2 observations, respectively. Sampling distribution of is assumed to be normally distributed. A linear combination of normally distributed random variables is also normally distributed. If underlying distribution is not normal, then by the central limit theorem, the sampling distribution of is approximately normal only if both n1 > 30 and n2 > 30. Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.1 Inference Concerning the Difference Between Two Means LO 10.1 If s21 and s22 are known, a 100(1  a)% confidence interval of the difference between two population means m1  m2 is given by Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.1 Inference Concerning the Difference Between Two Means LO 10.1 If s21 and s22 are unknown but assumed equal, a 100(1  a)% confidence interval of the difference between two population means m1  m2 is given by where and and are the corresponding sample variances and Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.1 Inference Concerning the Difference Between Two Means LO 10.1 If s21 and s22 are unknown but cannot be assumed to be equal, a 100(1  a)% confidence interval of the difference between two population means m1  m2 is given by where Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.1 Inference Concerning the Difference Between Two Means LO 10.1 Hypothesis Test for m1  m2 When conducting hypothesis tests concerning m1  m2 , the competing hypotheses will take one of the following forms: where d0 is the hypothesized difference between m1 and m2. Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.1 Inference Concerning the Difference Between Two Means LO 10.1 Test Statistic for Testing m1  m2 when the sampling distribution for is normal. If s21 and s22 are known, then the test statistic is assumed to follow the z distribution and its value is calculated as Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.1 Inference Concerning the Difference Between Two Means LO 10.1 Test Statistic for Testing m1  m2 when the sampling distribution for is normal. If s21 and s22 are unknown but assumed equal, then the test statistic is assumed to follow the tdf distribution and its value is calculated as where and Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.1 Inference Concerning the Difference Between Two Means LO 10.1 Test Statistic for Testing m1  m2 when the sampling distribution for is normal. If s21 and s22 are unknown and cannot be assumed equal, then the test statistic is assumed to follow the tdf distribution and its value is calculated as: where is rounded down to the nearest integer. Statistical Inference Concerning Two Populations

10.2 Inference Concerning Mean Differences LO 10.2 Make inferences about the mean difference based on matched-pairs sampling. Matched-Pairs Sampling Parameter of interest is the mean difference D where D = X1  X2 , and the random variables X1 and X2 are matched in a pair. Both X1 and X2 are normally distributed or n > 30. For example, assess the benefits of a new medical treatment by evaluating the same patients before (X1) and after (X2) the treatment. Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.2 Inference Concerning Mean Differences LO 10.2 Confidence Interval for mD A 100(1  a)% confidence interval of the mean difference mD is given by where and sD are the mean and the standard deviation, respectively, of the n sample differences, and df = n  1. Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.2 Inference Concerning Mean Differences LO 10.2 Hypothesis Test for mD When conducting hypothesis tests concerning mD, the competing hypotheses will take one of the following forms: where d0 typically is equal to 0. Statistical Inference Concerning Two Populations

Statistical Inference Concerning Two Populations 10.2 Inference Concerning Mean Differences LO 10.2 Test Statistic for Hypothesis Tests About mD The test statistic for hypothesis tests about mD is assumed to follow the tdf distribution with df = n  1, and its value is where and sD are the mean and standard deviation, respectively, of the n sample differences, and d0 is a given hypothesized mean difference. Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.3 Discuss features of the F distribution Inferences about the ratio of two population variances are based on the ratio of the corresponding sample variances . These inferences are based on a new distribution: the F distribution. It is common to use the notation F(df1,df2) when referring to the F distribution. Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.3 10.3 Inference Concerning Differences among Many Means The distribution of the ratio of the sample variances is the F(df1,df2) distribution. Since the F(df1,df2) distribution is a family of distributions, each one is defined by two degrees of freedom parameters, one for the numerator and one for the denominator Here df1 = (n1 – 1) and df2 = (n2 – 1). Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.3 10.3 Inference Concerning Differences among Many Means Fa,(df1, df2) represents a value such that the area in the right tail of the distribution is a With two df parameters, F tables occupy several pages. Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.4 Make inferences about the difference between three or more population means using an analysis of variance (ANOVA) test. Analysis of Variance (ANOVA) is used to determine if there are differences among three or more populations. One-way ANOVA compares population means based on one categorical variable. We utilize a completely randomized design, comparing sample means computed for each treatment to test whether the population means differ. Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.4 10.3 Inference Concerning Differences among Many Means The competing hypotheses for the one-way ANOVA: H0: µ1 = µ2 = … = µc HA: Not all population means are equal Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.4 10.3 Inference Concerning Differences among Many Means We first compute the amount of variability between the sample means. Then we measure how much variability there is within each sample. A ratio of the first quantity to the second forms our test statistic, which follows the F(df1,df2) distribution Between-sample variability is measured with the Mean Square for Treatments (MSTR) Within-sample variability is measured with the Mean Square for Error (MSE) Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.4 10.3 Inference Concerning Differences among Many Means Calculating MSTR 1. 2. 3. Calculating MSE 1. 2. Statistical Inference Concerning Two Populations

10.3 Inference Concerning Differences among Many Means LO 10.4 10.3 Inference Concerning Differences among Many Means Given H0 and HA (and a) the test statistic is calculated as where df1 = (c-1) and df2 = (nT-c) Statistical Inference Concerning Two Populations