Central limit theorem Let’s work with it.

Mix and match Sample Census Target population Statistic Parameter A complete collection of items desired to be studied A list of all of the items in the population A subset of a larger collection of items A sample from a homogeneous subset of the population A characteristic of a sample Occurs if a sampling method distorts a property of the population A comprehensive study of every item in a population The result if a respondent chooses not to answer questions A characteristic of a population Sample chosen so that all subsets of size n are equally likely Sample Census Target population Statistic Parameter Sampling frame Simple random sample Stratified sample Bias nonresponse

Central Limit Theorem What is the CLT? When can we use it?

What is the law of large numbers? How does it work?

Example Pepsi fills its 20 oz bottles with a machine which fills each bottle running on a normal distribution. The machine fills with an average of 20 oz and a standard deviation of 1.2 oz. Sometimes, the machine malfunctions and starts overfilling all the bottles and making quite the mess. (a) Can you make a judgement on one random bottle as to the amount of soda in it? Why? (b) If the machine starts malfunctioning, the average changes to 23oz. What is the average of the sampling distribution for 𝑥 .

Example Let’s say that the machine fills on a non normal distribution with a mean of 20 oz and a standard deviation of 1.2 oz. (a) Can you make a judgement on one random bottle as to the amount of soda in it? Why? (b) If the machine starts malfunctioning, the average changes to 23oz. What is the average of the sampling distribution for samples of 30 bottles ( 𝑥 ).

So, say we don’t know the distribution the machine fills with So, say we don’t know the distribution the machine fills with. Because of CLT, we know sample averages will be normally distibuted. The machine fills with a mean of 20 oz and a standard deviation of 1.2 oz. Sometimes, the machine malfunctions and starts overfilling all the bottles and making quite the mess. You select a sample of 30 bottles. (a) How might you go about obtaining these 30 bottles (What kind of sampling is this) (b) What would be the 95th percentile of the sampling distribution? (c)What is the probability that your sample will have a mean between 18.8oz and 21.2 oz. (d) What is the probability that your sample will have a mean between 19.78oz and 20.22 oz. (e) If all 30 bottles in your sample are in the bottom 5% of filled bottles, what would be the highest average you could obtain? (f) Your actual average of the sample is 24.1 oz. Would you think the machine was working right?