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5.2 The Sampling Distribution of a Sample Mean One thing we’ve been saying (or hoping) is that a sample should have approximately the same statistical properties as the population. In particular, we would like the mean of a sample to be approximately the mean of our population. Given a random variable, one thing we can do is to look at a fixed sample size and write down a probability distribution of the mean. In this way, the mean itself is a random variable. The question is then: how does the probability distribution of the mean of a fixed sample size relate to the mean of the population? We would ask the same question of standard deviation. Central Limit Theorem The above questions and others are answered by the Central Limit Theorem: Let x represent data values from a population with mean μ, and let x represent the sample mean defined for random samples of size n. Then x is a random variable with the following properties. 1. The mean of x is μ. 2. The standard deviation of x is σ/n1/2. 3. x will be approximately normal when n is sufficiently large (n greater than 30). The last part of the theorem tells us that even if x is not normal, x is closer and closer to normal for larger values of n. This may be offered as one reason why so many variables in the real world are approximately normal. If your random variable can be considered a mean, then it is approximately normal. Examples At a large factory, the mean wage is $42,500 and the standard deviation is $2000. What is the probability that the mean wage of 75 randomly selected workers will exceed $43,000? A biology teacher had noted that the scores on a standard exam attained by students from her past classes had a mean of 74 and a standard deviation of 14. The teacher decided to use a new book. Using the new book, a class of 50 students scored a mean of 78 on the standardized exam. Find the probability that a class of 50 students using the new book would have a sample mean as large or larger than 78 if the new book were equivalent to the old book. Is this evidence to support the teacher’s claim that the new book is superior? We are now beginning to hint at the idea of hypothesis testing and p-values. Practice Problem The scores of high school seniors on the ACT in 2003 had mean μ=20.8 and σ=4.8. The distributions of scores is not normal. 1. What is the approximate probability that a single random student chosen from all those taking the test scores 23 or higher? 2. What are the mean and standard deviation of the sample mean score of a SRS of 25 students? 3. What is the approximate probability that the mean scores of these students is 23 or higher? 4. Which of your two Normal probability calculations is more accurate? Why?