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chapter 4-Sampling Distribution

chapter 4-Sampling Distribution - CHAPTER 4 SAMPLES AND...

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CHAPTER 4 SAMPLES AND SAMPLING DISTRIBUTIONS 1. Why do We Use Samples? 2. Probability Sampling 2.1. Simple Random Samples 3. Sampling Distributions 3.1. The Sampling Distribution of the Sample Mean x ̄ 3.1.1. The Expected Value of x ̄ 3.1.1.1. The Relationship Between the Mean of the Parent Population and the Mean of All x ̄ Values 3.1.2. The Variance of x ̄ and Standard Error of x ̄ 3.1.3. The Relationship Between the Variance of the Parent Population and the Variance of x ̄ 3.1.4. The Shape of the Sampling Distribution of x ̄ . The Relationship Between the Parent Population Distribution and the Sampling Distribution 3.1.5. Examples Using the Normal Sampling Distribution of x ̄ 3.1.6. The Margin of Sampling Error (MOE) 3.1.7. Error Probability α 3.1.8. Determining the Sample Size for a Given MOE 3.2. The Sampling Distribution of the Sample Proportion p ̄ 3.2.1. The Expected Value of p ̄ 3.2.1.1. The Relationship Between the Parent Population Proportion and the Mean of All p ̄ Values 3.2.2. The Variance of p ̄ and Standard Error of p ̄ 3.2.2.1. The Relationship Between Variance of the Binary Parent Population and the Variance of p ̄ 3.2.3. The Sampling Distribution of p ̄ as a Normal Distribution 3.2.4. Margin of Error for p ̄ 3.2.5. Determining the Sample Size for a Given MOE 1. Why do We Use Samples? Sampling is the basis of inferential statistics. A sample is a segment of a population. It is, therefore, expected to reflect the population. By studying the characteristics of the sample one can make inferences about the population. There are several reasons why we take a part of the population to study rather than taking a full census of the population. These are: Samples cost less. Sampling takes less time. Samples are more accurate. Sample observations are usually of higher quality because they are better screened for errors in measurement and for duplication and misclassifications; Samples can be destroyed to gain information about quality (destructive sampling). 2. Probability Sampling A sample in which each element of the population has a known and nonzero chance of being selected is called a probability sample. 2.1. Simple Random Samples Chapter 4—Sampling Distributions Page 1 of 32
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A simple random sample is a probability sample in which all possible samples of size n are equally likely to be chosen. To explain this requirement, let the population consist of letters A, B, C, D, and E. Since there are five items in the population, then N = 5. We want to select a sample of size 3, that is, n = 3. Since sampling is random (the letters are written on little balls and are put in a bowl), there is more than one way that we can select 3 items from 5 items. Using the combination formula, the total number of possible samples is C( N , n ) = C(5, 3) = 10. The following is the list of all 10 possible samples: ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE The definition of SRS implies that each sample has the equal chance of 0.10 of being selected. This process of simple random selection applies to a finite (small) population. The simple random selection process is different when the population is infinite (large).
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