The Central Limit Theorem

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Dulcie Payne
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1 - The Central Limit Theorem Definition Sampling Distribution of the Mean the probability distribution of sample means, with all samples having the same sample size n. (In general, the sampling distribution of any statistic is the probability distribution of that statistic.) Given: Central Limit Theorem. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation σ.. Simple random samples all of the same size n are selected from the population. (The samples are selected so that all possible samples of size n have the same chance of being selected.)

2 Central Limit Theorem Conclusions:. The distribution of sample means x will, as the sample size increases, approach a normal distribution.. The mean of the sample means will be the population mean µ.. The standard deviation of the sample means will approach σ/ n. Practical Rules Commonly Used:. For samples of size n larger than, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets better as the sample size n becomes larger.. If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size n (not just the values of n larger than ). Notation the mean of the sample means µ x = µ the standard deviation of sample means σ x = σ n (often called standard error of the mean)

3 SSN digits x Distribution of digits from Social Security Numbers (Last digits from students) Frequency Distribution of digits Distribution of Sample Means for Students Frequency

4 As the sample size increases, the sampling distribution of sample means approaches a normal distribution. distributed weights with a mean of lb and a standard deviation of lb, that his weight is greater than lb. b) if different men are randomly selected, find the probability that their mean weight is greater than lb. distributed weights with a mean of lb. and a standard deviation of lb, that his weight is greater than lb. z = =.

distributed weights with a mean of lb. and a standard deviation of lb, that his weight is greater than lb. The probability that one man randomly selected has a weight greater than lb. is. distributed weights with a mean of lb.

5 distributed weights with a mean of lb. and a standard deviation of lb, that his weight is greater than lb. The probability that one man randomly selected has a weight greater than lb. is. distributed weights with a mean of lb. and a standard deviation of lb, b.) if different men are randomly selected, find the probability that their mean weight is greater than lb. z = =. distributed weights with a mean of lb. and a standard deviation of lb, b.) if different men are randomly selected, find the probability that their mean weight is greater than lb. The probability that the mean weight of randomly selected men is greater than lb. is..

6 distributed weights with a mean of lb and a standard deviation of lb, that his weight is greater than lb. P(x > ) =. b) if different men are randomly selected, their mean weight is greater than lb. P(x > ) =. It is much easier for an individual to deviate from the mean than it is for a group of to deviate from the mean. Sampling Without Replacement If n >. N N - n σ x = σ n N - finite population correction factor Example: IQ scores are normally distributed and have a mean of and a standard deviation of. If people are randomly selected, find the probability that their mean is at least.

Probability and Samples. Sampling. Point Estimates

Probability and Samples. Sampling. Point Estimates Probability and Samples Sampling We want the results from our sample to be true for the population and not just the sample But our sample may or may not be representative of the population Sampling error