Sampling Distributions & the Central Limit Theorem

2. Sampling Distributions & the Central Limit Theorem A sampling distribution is a distribution consisting of a particular sample statistic from a collection of samples of the same size (n) that are repeatedly taken from a population. If the mean is the sample statistic, the distribution is called the sampling distribution of sample means. Multiple samples are taken from the same population and the mean is recorded for each sample. Each mean would be indicated as . Population: m, s S5 S2 S4 S1 S3

3. The Central Limit Theorem • Describes the relationship between the sampling distribution and the population from which the samples were taken. • If samples of size n, where n ≥ 30, are taken from ANY population with a mean of m and standard deviation of s, then the sample distribution of sample means approximates anormal distribution (bell curve). The larger the sample size, the better the approximation. • If the population is already normally distributed, then a sampling distribution of sample means is normally distributed for ANY sample size, n.

4. Properties of Sampling Distributions of Sample Means: • The mean of the sample means, , is equal to the population mean, m. • 2) The standard deviation of the sample means, , is equal to the population standard deviation, s, divided by the square root of the sample size. This is called the standard error of the mean.

5. If either case is true, then the following relationships apply:

6. Example: Phone bills for residents of Cincinnati have a mean of $64 and a standard deviation of 49, as shown in the graph below. Random samples of 36 phone bills are drawn from this population and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. What is the probability that a randomly selected sample of 36 phone bills will have a mean that falls between $50 and $75?

7. Probability and the Central Limit Theorem Distribution must be approximately normally distributed and this is to find the probability for the mean of a sample. • z-score method: • Convert desired mean value(s) to a z-score: • Look up z-score(s) on table for the probability • OR • Use normalcdf function with m = 0 and s = 1 • Calculator Method: • Find sampling distribution mean, mx and the standard error, • Use normalcdf function with sampling mean and standard error

8. Credit card balances are normally distributed, with a mean of $2870 and a standard deviation of $900. • What is the probability that a randomly selected credit card holder has a balance less than $2500? • b) If you randomly select 25 credit card holders, what is the probability that their mean credit card balance is less than $2500?

9. Which is more likely? The heights of American women aged 20 – 29 are normally distributed with a mean of 64 inches and a standard deviation of 2.4 inches. Are you more likely to select randomly one woman with a height less than 70 inches or are you more likely to select a sample of 20 women with an average height less than 70 inches?