1. Stat 321 – Day 20 Confidence Intervals (7.1)

2. A small fire breaks out in the dean’s office in a waste basket. A physicist, a chemist and a statistician come running to the rescue. An argument ensues: The physicist argues that they should measure the heat of the fire, then apply sufficient cold to lower the temperature below the ignition point. The chemist argues they should assay the material in the wastebasket and identify a suitable reagent to neutralize the oxidation reaction. The statistician, meanwhile, has been off setting fire to all the other wastebaskets in the office…. When confronted as to why he would do such a thing, he replied, “To get a large enough sample size, of course!”

3. Recap Random samples: Getting samples that are truly representative of the population takes care (unbiased sampling method) Sampling Distribution: Distribution of sample statistics in repeated random samples from the same population have a predictable pattern  Ideally centered at parameter of interest  Often larger samples => less variability

4. Deriving Sampling Distributions Exact Sampling Distributions  Figure out all the values the statistic can take and find the probability of each  Only for very small n! Central Limit Theorem: Sampling distribution is (approx) normal with known mean and std dev  Only applies to averages and large n or normal population  n may need to be pretty large w/ discrete parent distribution Simulation  Randomly select many, many samples (enough to see the long-term pattern) and calculate the statistic for each sample, examine average and SD of the statistics  Especially useful when n is “medium-sized” or the statistic is not a mean

5. Examples Population Normal(20, 8) Sampling Distribution of median Sampling Distribution of range Population range = 54

6. Last Time – Central Limit Theorem The sampling distribution of the sample mean is normal (if the population is normal) or approximately normal (if the sample size is large) with expected value equal to  and standard deviation equal to  /  n Allows us to calculate probabilities such as P( > 50) when know  and 

7. Example 1: Candy Bar Weights

9. Example 1: Candy Bars 95% of sample means should fall within two standard deviations of the population mean (empirical rule) In 95% of samples, statistic is within 2 standard deviations of the population mean In 95% of samples, population mean is within 2 standard deviations of the statistic The interval  2  should contain 

10. Example 2 agemom.mtw For a population of 1199 mothers, recorded their age when gave birth to first child. Describe distribution:  Positive skewed   =average age in population = 22.519 years   =standard deviation in population = 4.885 years

11. Confidence Interval We refer to these as 95% confidence intervals because if we were to repeatedly take random samples from the population and construct intervals in this manner, then in the long run 95% of those intervals would succeed at including the value of the population mean . When we apply this to one sample, we can think of this as a statement of how reliable our method is.

12. Next General form: estimate + margin of error What if we wanted to be 68% confident? 99% confident? When is this procedure valid? What affects the reliability and precision of our estimate? Quiz Thursday on HW 6