Spring 2012 Chap 5

Spring 2012 Chap 5 - Chapter 5Normal Distributions with...

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Chapter 5—Normal Distributions with Sample Means (sec. 5.2) In section 1.3, we learned how to use the normal distribution for X if we knew the μ and σ AND if X was normally distributed. Now, we will see how to use the normal distribution for sample means. Normal distribution for X What is X ? X is a single observation of a variable we are interested in (i.e. a score or account balance or height or weight or grade). When do we use normal distribution? Use if variable X is normally distributed. Mean and standard deviation of X: μ and σ (told to us in the problem) Formula for Z: To be able to use tables to determine probabilities, we standardize (i.e.convert) ******************************* In Chapter 3--we were looking at the POPULATION DISTRIBUTION of a variable, X Population distribution: the distribution of the values X takes for all members of the population. The population distribution is also the probability distribution of any variable, X , when we choose one individual observation from the population at random. Now, we want to find out about the distribution of sample statistics , not just the distribution of individual values. The Distribution of a Statistic: A statistic from a random sample or randomized experiment is a random variable. The probability distribution of the statistic is its SAMPLING DISTRIBUTION We will look at the sampling distribution for the sample mean The Sampling Distribution of a Sample Mean STAT 301 Spring 2012 Chapter 5 Page 1
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is the sample mean If the original observations, X, are from a normal distribution, then the distribution of is also normally distributed. That is, if the population has a normal distribution, so does the sampling distribution. Then we can proceed as in chapter 1. So, what if the population distribution is not a normal distribution or of unknown type? To deal with these other distributions, we have…………… THE CENTRAL LIMIT THEOREM!!!! Slide above was used with permission from the Freedman’s website. The Central Limit Theorem says that when your sample size ( n ) is large enough, the distribution of will be normally distributed regardless of the original distribution of X . THIS IS A VERY IMPORTANT CONCEPT!!!! is approximately Where: is the mean of the distribution of and is the standard deviation of . Note: The formula we use for Z: Example 1 : The scores of students on the ACT college entrance examination in 2001 had mean and standard deviation. The distribution of scores is only roughly normal. STAT 301 Spring 2012 Chapter 5 Page 2
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a. What is the approximate probability that a single student randomly chosen from all those taking the test scores is 23 or higher? b. Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the sample mean score of these 50 students? c. What is the distribution of for the 50 students?
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