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Download Chapter 7, part C
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Chapter 7, part C VI. The Central Limit Theorem We will use the CLT to determine the form of the probability distribution of x There are two cases, when the population distribution is unknown (the most likely) and when it is known. A. The Theorem “In selecting simple random samples of size n from a population, the sampling distribution of x can be approximated by a normal probability distribution as the sample sizes become large.” As a rule of thumb, if n30, the sample satisfies the CLT. Try this link for an example of how the CLT works with simply tossing a coin. Usefulness The beauty of the CLT is that even if you have no idea the form of the probability distribution of the overall population, sampling in large sizes will give you a sampling distribution that is normal. Then you can calculate probabilities like we’ve already done. Remember the SPEND example? This portrays the distribution of the population and it hardly looks like a normal distribution, does it? Eric R. Dodge: thanks to John Ottensman at IUPUI. By taking small samples, 500 times, the distribution begins to look normal. But the CLT requires large samples. Eric R. Dodge: thanks to John Ottensman at IUPUI. Taking more samples fills in some of the gaps, but we’re still using a small sample. Watch what happens to the distribution when we take larger samples. Eric R. Dodge: thanks to John Ottensman at IUPUI. Mean = 41.48, standard error = 8.13 minutes. Eric R. Dodge: thanks to John Ottensman at IUPUI. Mean = 41.34, standard error = 4.58 minutes. We’ll talk more about how increasing n changes things in a few slides. Eric R. Dodge: thanks to John Ottensman at IUPUI. B. The Value of the Sampling Distribution of x When we take a sample, we calculate x , which will almost certainly not be equal to . Sampling error = x The practical value is to use the sampling distribution to provide probability information about the size of the sampling error. EAI Example The personnel director believes a sample is an acceptable estimate of if: x x $500 So we need to find the probability of a sampling error being no more than $500. Probabilities If we find this area, we just multiply by 2. x 730.30 51,300 E( x)=51,800 52,300 x Calculating Probabilities Find the z-score: z=(52,300-51,800)/730.3 = .68 Area from the standard normal table, z=.68 is .2518 so P x $500 50.36% C. Sample Size and Sampling Distribution There’s little more than a 50% chance that the sampling error will be less than $500 in our previous example. One way to improve our odds is to take a larger sample. As you saw in the earlier “SPEND” slides, the only thing that changes is the standard error of the sampling distribution. The standard error decreases Previously, 4000 x ( ) 730.30 n 30 but if we take a sample size of n=100, 4000 x ( ) 400 n 100 and the sampling distribution gets more narrow, just like the SPEND examples when we increased n from 8 to 25 to n=75. And the probability increases Find the z-score: z=(52,300-51,800)/400 = 1.25 Area from the standard normal table, z=1.25 is .3944 so P x $500 78.88% A larger sample size will provide a higher probability that the value of x is within a specified distance of .
