Stat 226 - Section 4.4 - Preliminary Terminology Random...

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Chapter 4 Probability and Sampling Distributions Chapter 4 1 Preliminary Terminology Random behavior is unpredictable in the short term but has a regular pattern in the long run that follows the laws of probability. The outcomes of independent events are not influenced by any others. We aim to have observations on different individuals be independent. A random variable is a variable whose value is a numerical outcome of a random event. A statistic (such as the mean) calculated from a randomly chosen sample is an example of a random variable. Random variables typically have a distribution , which is the set of values the random variable can take and how often it takes each value. Chapter 4 2 Section 4.4 The Sampling Distribution of a Sample Mean Section 4.4 3 Sampling Variability In section 3.3, we saw that a statistic from a random sample will take different values if we take more samples from the same population. The statistic is a random variable and varies according to its sampling distribution . In this section we will examine the sampling distribution of the sample mean ¯ x . Section 4.4 4 Sampling Distribution Defined The sampling distribution of a statistic is the set of values that the statistic could take under all possible samples of size n from the population and how frequently the statistic takes each value. Section 4.4 5 Estimation Why do we compute ¯ x for a sample in the first place? If we would like to know something about the population mean μ , it seems ¯ x would be a reasonable estimate of μ . A simple random sample (SRS) should “fairly” represent the population, so the mean ¯ x should (on average) be near the mean μ of the population. By “fairly”, we mean that a SRS reduces the chance of bias. The mean ¯ x from a SRS is an unbiased estimate of μ . Section 4.4 6
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Sampling Variability Will ¯ x be exactly equal to μ ? We do not know μ , so there is no way to know for sure. Most likely, ¯ x is not exactly equal to μ . Recall that ¯ x varies from sample to sample. If we choose another SRS, we will probably get a different value of ¯ x . With all of this variability, why is ¯ x a reasonable estimate of μ ? Section 4.4 7 Sampling Variability ¯ x is a reasonable estimate of μ because of the characteristics we saw in section 3.3. As we include more and more individuals in our sample, probability theory assures that the statistic ¯ x will get closer and closer to the population parameter μ . The variability or spread of the sampling distribution becomes smaller.
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