ComparingTwoPopulations-solutionsPart-a

# ComparingTwoPopulations-solutionsPart-a - Comparing two...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 = radicalBigg ( 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 . 0 Assume that brand A battery lifetimes: x 1 , . . . , x 5 are a realization of a random sample from N ( μ 1 , σ ) and brand B battery lifetimes: y 1 , . . . , y 6 are a realization of an independent random sample from N ( μ 2 , σ ). Using the data, compute estimates for μ 1 - μ 2 , and σ .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern