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Ch12 - Chapter 12 Inference About One Population 1 12.1...

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2 12.1 Introduction 12.1 Introduction In this chapter we utilize the approach developed In this chapter we utilize the approach developed before to describe a population. before to describe a population. Identify the parameter to be estimated or tested. Identify the parameter to be estimated or tested. Specify the parameter’s estimator and its sampling Specify the parameter’s estimator and its sampling distribution. distribution. Construct a confidence interval estimator or perform Construct a confidence interval estimator or perform a hypothesis test. a hypothesis test.
3 We shall develop techniques to estimate and test three population parameters. Population mean μ Population variance σ 2 Population proportion p 12.1 Introduction 12.1 Introduction

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4 Recall that when σ is known we use the following statistic to estimate and test a population mean When σ is unknown, we use its point estimator s, and the z-statistic is replaced then by the t-statistic 12.2 Inference About a Population Mean 12.2 Inference About a Population Mean When the Population Standard Deviation When the Population Standard Deviation Is Unknown Is Unknown n x z σ μ - =
5 The t - Statistic The t - Statistic n x μ - = s n x σ μ - = Z t σ σ σ σ s s s s When the sampled population is normally distributed, the t statistic is Student t distributed. Z Z Z ZZ t t t t t t t t t s s s s s σ σ σ σ σ t

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6 The t - Statistic The t - Statistic n x μ - = s 0 The t distribution is mound-shaped, and symmetrical around zero. The “degrees of freedom”, (a function of the sample size) determine how spread the distribution is (compared to the normal distribution) d.f. = v 2 d.f. = v 1 v 1 < v 2 t Using the t-table
7 Testing Testing μ μ when when σ σ is unknown is unknown Example 12.1 - Productivity of newly hired Trainees

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8 Example 12.1 In order to determine the number of workers required to meet demand, the productivity of newly hired trainees is studied. It is believed that trainees can process and distribute more than 450 packages per hour within one week of hiring. Can we conclude that this belief is correct, based on productivity observation of 50 trainees (see file Xm12-01 ). Testing Testing μ μ when when σ σ is unknown is unknown
9 Example 12.1 – Solution The problem objective is to describe the population of the number of packages processed in one hour. The data are interval. H 0 : μ = 450 H 1 : μ > 450 The t statistic d.f. = n - 1 = 49 n s x t μ - = We want to pr ove that the trainees reach 90% productivity of experienced workers We want to pr ove that the trainees reach 90% productivity of experienced worker s Testing Testing μ μ when when σ σ is unknown is unknown

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10 Solution continued (solving by hand) The rejection region is t > t α ,n – 1 t α ,n - 1 = t .05,49 2245 t .05,50 = 1.676.
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Ch12 - Chapter 12 Inference About One Population 1 12.1...

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