1. The Central Limit Theorem Section 9.3.2

2. Starter Assume I have 1000 pennies in a jar Let X = the age of a penny in years –If the date is 2007, X = 0 –If the date is 2006, X = 1, etc Draw a smooth curve that is your best guess of the shape of the distribution of X Write a verbal description of your guess of the distribution

3. Today’s Objectives Perform an activity that demonstrates the effect of the Central Limit Theorem Write a statement of The Central Limit Theorem California Standard 9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.

4. How Old Are Pennies in Circulation? That’s a hard question to answer. Let’s try an easier one: What’s the average age of 1000 randomly collected pennies? –We could agree that the answer to this question is a reasonable approximation to the answer to the main question. We could now actually record the ages of all 1000 pennies and calculate the mean, or we could approach the problem by sampling. I will give you 25 pennies in a cup that are a sample of the 1000 pennies. We will take samples of increasing size and see what happens.

5. Sample the population n = 2 Shake up your pennies in the cup and select a sample of 2 pennies Calculate the mean age of the 2 pennies –Record the sample mean you found Replace the pennies, shake again, and choose another sample of 2 pennies –Record the new sample mean Plot your sample means on the whiteboard –You should be plotting two points Draw a sketch and write a description of the sampling distribution

6. Sample the population n = 10 Shake up your pennies in the cup and select a sample of 10 pennies Calculate the mean age of the 10 pennies –Record the sample mean you found Replace the pennies, shake again, and choose another sample of 10 pennies –Record the new sample mean Plot your sample means on the whiteboard –You should be plotting two points Draw a sketch and write a description of the sampling distribution

7. Sample the population n = 25 Combine your pennies with one or more partners and draw a new sample of 25. Record the ages of all 25 of your pennies and calculate the sample mean. Plot your sample mean on the whiteboard. Draw a sketch and write a description of the sampling distribution. –NOTE: KEEP YOUR SAMPLE MEAN IN YOUR NOTES. WE WILL USE IT AGAIN IN CHAPTER 10.

8. The Central Limit Theorem If an SRS of size n is drawn from a population of any shape with mean μ and standard deviation σ, and if n is large, then the sampling distribution of the sample means will be approximately normal with mean μ and standard deviation σ/ √n –How large is large? For most situations, n=30 will be large enough. –The more “non-normal” the population, the larger n should be to get good results

9. Today’s Objectives Perform an activity that demonstrates the effect of the Central Limit Theorem Write a statement of The Central Limit Theorem California Standard 9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.

10. Homework Read pages 487 - 491 Do problems 30 - 34