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Unformatted text preview: Comparing two populations or processes 1 Introduction • In previous chapters, we introduced models for a characteristic of units in a population, models for a process, and developed procedures (confidence intervals & hypothesis tests) to make inference for the parameters (e.g. μ & θ ) of these models using data. • In this chapter, we will compare a characteristic of the units in two populations: – This characteristic of the units in population 1 is modeled with a distribution with mean μ 1 and standard deviation σ 1 . – This characteristic of the units in population 2 is modeled with a distribution with mean μ 2 and standard deviation σ 2 . • We will introduce procedures (confidence intervals & hypothesis tests) to make infer- ence for the difference: μ 1- μ 2 , which itself is a parameter, using data. (e.g. is there statistical evidence that μ 1- μ 2 > 0). • These methods also apply to the comparison of two processes. • Procedures to compare σ 1 and σ 2 will be introduced in a supplement available on Moodle. 2 Unpaired comparisons: the two independent sam- ples t-test and confidence interval for μ 1- μ 2 Examples of unpaired comparisons of means • Compare the mean starting salary of graduates of UMN–TC with graduates from Ohio State. • Compare the mean lifetime of Duracell Batteries with that of Energizer batteries. • Compare the mean LDL cholesterol reduction of those given a new treatment (treated) with those given the present treatment (control). 1 2.1 Two independent samples from two Normal distributions • Population/Process 1’s sample measurements x 1 ,...,x n 1 are a realization of a random sample of size n 1 from N ( μ 1 ,σ ). Let ¯ x & s 1 denote the observed sample mean & standard deviation. • Population/Process 2’s sample measurements y 1 ,...,y n 2 are a realization of a random sample of size n 2 from N ( μ 2 ,σ ). Let ¯ y & s 2 denote the observed sample mean & standard deviation. • The two random samples are independent . • The quantity ¯ x- ¯ y is an estimate of μ 1- μ 2 . • The quantity s p , defined as: s p = s ( n 1- 1) s 2 1 + ( n 2- 1) s 2 2 n 1 + n 2- 2 is an estimate of σ . This is a realization of the estimator S p . We are assuming that Population/Process 1’s distribution has the same standard deviation σ as Population/Process 2’s distribution. Example 2.1: Suppose we wish to compare the mean lifetimes (in hours) of two types of batteries (Brand A and Brand B). We acquire 5 brand A batteries and measure lifetimes in a portable CD player: 10 . 5 , 9 . 6 , 10 . 2 , 9 . 9 , 10 . 4 We also acquire 6 brand B batteries and measure lifetimes in a portable CD player: 8 . 9 , 9 . 4 , 9 . 7 , 9 . 3 , 8 . 8 , 10 ....
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This note was uploaded on 05/06/2011 for the course STAT 5021 taught by Professor Staff during the Spring '08 term at Minnesota.
- Spring '08