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STA3032 Chapter 5 and 6 Inference for two populations

STA3032 Chapter 5 and 6 Inference for two populations -...

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Comparing Two Population parameters Some new concepts The framework: Until now we had one population, one random sample from that population and just one parameter (μ or π) with an unknown value and we made inference about the unknown value of the parameter. In this Chapter We have two populations, a population of X’s [say, {X 1 , X 2 , …, X N } ] and a population of Y’s, [say, {Y 1 , Y 2 , …, Y M }]. If X and Y are both quantitative variables, the population means are μ X and μ Y , and standard deviations are σ X and σ Y , respectively; we are interested in making inferences about the difference of population means, μ X – μ Y . If both X and Y are categorical variables, each with two categories, then the parameter of interest is the difference between the proportion of “Success”s, denoted by π X , in the population of X’s and proportion of “Success”s, denoted by π Y in the population of Y’s. We select (simple) random sample of size n X from the population of X’s and a (simple) random sample of size n Y from the population of Y’s to make inferences about the difference of the parameters of interest in the respective populations. The samples can be either independent of each other or they may be dependent on each other. Definition: Two random samples are said to be independent samples if the selection of a unit from one population has no effect on the selection or non-selection of another unit from the second population. Otherwise the samples are said to be dependent samples. Independent samples are used in most applications. However, in some applications the selection of one unit will from one of the populations determines the selection of another one from the second population. Such samples are said to be dependent samples. In such applications one unit from each population come in pairs. Thus we have a random sample of pairs . These pairs either come naturally (e.g., twins-studies, studies of married couples, observations on the same person under two different conditions, etc.) or the pairs are created by the experimenter, being matched on as many characteristics as possible, except the one characteristic of interest to the researcher. We have the usual general formulas for confidence intervals and the usual 6 steps of making statistical tests. STA 3032 Chap 6 Part 2, Page 1 of 20

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Comparing The Means Of Two Population Using Independent Samples (Sections 5.4, 5.6, 6.5, 6.7) Notation: Suppose we have a population of X’s with mean μ X and standard deviation σ X and a population of Y’s with mean μ Y and standard deviation σ Y . We are interested in the difference if the population means, μ X μ Y . We select a simple random sample, of size n X , from the population of X’s and calculate that sample statistics, (i.e., X and S X ). Similarly we select an independent simple random sample of size n Y from the population of Y’s and calculate Y and S Y .
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