1. Distribution of Sample Means 2011, 10, 20

2. Today ’ s Topics What is distribution of sample means?** Properties of distribution of sample means* How to find out the probability of a sample (mean)?**

3. The sales of food and drink in Jamie Joan’s vary from day to day. The distribution of daily sales is a normal distribution with mean of  = $900 and a standard deviation of  = $300. We plot all weekly mean sales Where would the weekly means fluctuate around? How much variability would you expect in the distribution of weekly means? Example 1: Jamie Joan ’ s ― Weekly Mean Distribution of Weekly Means

4. Distribution of Sample Means If the sample size is large (n > 30) PopulationDistribution of sample means “Good ”

5. Properties of distribution of sample means – Central Limit Theorem For any population with mean  and standard deviation , the distribution of sample means for sample size n will approach a normal distribution with a mean of  and a standard deviation of as n approaches infinity. (good approximation if n > 30). Mean: Standard Error:

6. The sales of food and drink in Jamie Joan’s vary from day to day. The distribution of daily sales has mean of  = $900 and a standard deviation of  = $300. Questions What is the probability that a weekly mean is between $560 and $1,240? What is the probability that a weekly mean is at or greater than $900? The mean sales for last week is $1,100. Did they have an exceptional week? (Hint: What is the probability of having a weekly mean of $1,100 or higher?) Example 1: Jamie Joan ’ s ― Weekly Means $560$1,240

7. Your Turn: IQ Score Population IQ: Distribution of Sample Means (n = 75) p = ?

8. How to Find out the Probability of a Sample Mean – Four Steps List the population parameters: Compute and list the mean and standard error of the distribution of sample means Compute the z score From the Unit Normal Table, look for the probability of z

9. Notations Sample X s Population   Distribution of sample means

10. Lecture Recap: Distribution of Sample Means What is a distribution of sample means? Central Limit Theorem (Properties of a distribution of sample means) Shape Mean Standard error How to use a distribution of sample means to find out the probability of a sample mean? Notations for population distribution, sample distribution, and distribution of sample means