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Distribution of the Sample Mean Page 1 The Central Limit Theorem The goal of this activity is to apply the Central Limit Theorem. The Central Limit Theorem Suppose that a simple random sample of size n is obtained from a large population with mean µ and standard deviation σ. The sampling distribution of x will have the following characteristics: 1. As the sample size increases, the sampling distribution will approach a normal distribution. 2. The mean of sample means is: µx = µ. 3. The standard deviation of the sample mean (called the standard error of the mean) is: σ σx = n 1. Nägele’s Rule (developed in the 1850’s) says the gestation period of humans is normally distributed with a mean of 266 days and a standard deviation of 16 days. A medical research team plans to study 10 pregnant women. The researchers want to compare their results with other such studies. a. Find the mean and standard deviation expected for comparison. b. Graph the population distribution and the sampling distribution in the same viewing window on your graphing calculator. Compare the two distributions. The following distribution function on your calculator may help. (Hint: use X as the value.) normalpdf(value,mean,standard deviation) c. Determine the probability that the women in the study will have a mean gestation period of 260 days or less. Robert A. Powers University of Northern Colorado Distribution of the Sample Mean Page 2 Practical Guidelines for the Central Limit Theorem If a random variable X is normally distributed, then the sampling distribution of x is normally distributed. If a random variable X is not normally distributed, but n > 30, then the sampling distribution of x is approximated reasonably well by a normally distribution, except under relatively rare circumstances. 2. Market analysts use a collection of stocks to determine trends. For example, the S & P 500 is a collection of 500 stocks of publicly traded companies whose values are used to determine monthly rates of return. A statistician thinks this is inefficient and proposes a new method called the R A P 20. This method starts with the current market value of stocks. Then randomly selects 20 stocks of publicly traded companies to determine their change in market value for the day and adjusts the market value of stocks accor The Central Limit Theorem for a Finite Population Suppose that a simple random sample of size n is obtained from a finite population of size N with mean µ and standard deviation σ. The sampling distribution of x will have the following characteristics: 1. As the sample size increases, the sampling distribution will approach a normal distribution. 2. The mean of sample means is: µx = µ. 3. The standard deviation of the sample mean is: N −n σ σx = ⋅ N −1 n Robert A. Powers University of Northern Colorado