Central Limit Theorem cHapter 18 part 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. 19-22) For quantitative data, we will look at sample means to create a model. (Means will be used in Ch. 23-25)

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.

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.

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

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

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 143.74 with a standard deviation of 51.54 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 = 51.54 200 =3.64 𝑧= 150−143.74 3.64 =1.72 With a z-score of 1.72, the sample mean weight of 150 is not unusually high.

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