1. Central Limit Theorem
cHapter 18 part 2

2. (Proportions will be used in Ch. 19-22)
For categorical data, we looked at sample proportions to create a model.
(Proportions will be used in Ch )
For quantitative data, we will look at sample means to create a model.
(Means will be used in Ch )

3. The Central Limit Theorem
The mean of a random sample is a random variable whose sampling distribution can be approximated by a Normal model. The larger the sample, the better the approximation will be.

4. CLT Conditions for modeling sample means:
Sampled values must be independent and samples must be randomly selected.
The sample size should be no more than 10% of the population.
The sample needs to be “large enough” yet there is no specific rule on this. Use your best judgment.

5. Notation 𝜇=𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛 𝑥 𝑜𝑟 𝑦 =𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛
𝑥 𝑜𝑟 𝑦 =𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛
𝜎=𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝜎 𝑦 𝑜𝑟 𝑆𝐷( 𝑦 )=𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

6. Sampling Distribution Model for a Mean
𝜇( 𝑦 )=𝜇
𝑆𝐷 𝑜𝑟 𝜎 𝑦 = 𝜎 𝑛

7. 𝑧= 150−143.74 3.64 =1.72 All conditions are met. 𝑥 𝑜𝑟 𝑦 =143.74
Example: The CDC reports that 18-year-old women have a mean weight of with a standard deviation of pounds. A random sample of 200 women reported a mean weight of 150 pounds. Is this an unusually high sample mean?
All conditions are met.
𝑥 𝑜𝑟 𝑦 =143.74
𝑆𝐷 𝑦 = 𝜎 𝑛 = =3.64
𝑧= 150− =1.72
With a z-score of 1.72, the sample mean weight of 150 is not unusually high.

8. Today’s Assignment:
You still need to read Chapter 18!
Add to HW: p.436 #37 & 38