Recap;

1. Recap; Chapter 6.2

2. Binomial Distribution: Basics: • Used with binomial categorical data. • Can be quantitative with two possible numerical responses, [0,1] • X is the number of 1’s in n trials • Sample size should be less than 10% of the population to insure independence. • npq should be at least 10, for the data to be normally distributed. • μ = np

3. Example; • A medication causes side effects in 35% of patients. Eight patients at our clinic are receiving the medication. Let X be the number of patients with side effects. • Find P(6). • How many of the eight would you expect to have side effects? • μ = np

4. How likely is a sample? Sec 8.2

5. Sample Distributions; • Descriptive statistics are random variables! • Means, Medians, Standard deviations, proportions, etc. Recall: a random variable was a variable for random phenomenon. • As Random Variables descriptive statistics have Probability Distributions. • Lets start by looking at proportions.

6. μ

7. In a population 35% of our subjects give a positive response. Population Distribution P(0)=0.65 and P(1)=0.35 Samples are taken which approximate the populations proportion Data Distributions P(0)=0.62 and P(1)=0.38 Typically those sample proportions will be near the population proportion, but some will be a little bit away and a few further out but almost all will be relatively close together. Sampling Distributions μ = p σ = μ

8. Standard Error; • In an effort to be clear we will refer to the standard deviation for a sample distribution as the standard error. Standard Error = Our understanding stays the same; Typically a sample will be within one standard error It would be unusual to find a sample outside of two standard errors Almost all (99.8%) of samples will fall within three standard errors.

9. But we can’t compute these!? • We will not normally know the true proportion. • There is no realistic way to find All possible samples of size n for all but the smallest of populations. • These are, for now, theoretical values. • There are some things we can do with them.

10. Example; • A local winery claims 75% of people living in the rogue valley prefer their wines. We want to put this claim to the test. • Since we can not ask everyone in the valley which wine they prefer we must take a sample. Say 100 people. • So we go out and sample 100 people and 60% of them say yes they do prefer that wine.

11. Example, • How likely were we to get this result assuming their claim or 75% was true. • If we look at a theoretical distribution of All samples of size 100 it should look like a normal distribution. With mean at 0.75. • We can calculate the standard error as; 0.75

12. Example, • Given that we should have a mean of .75 and a standard deviation of 0.0433, what is the probability that we would find a sample of 100 people for which only 60% said yes? • We know almost all results should be within 3 standard errors from the mean and thus in the interval; • So our 60% falls outside the interval in which 99.8% of all results should fall. 3(σ) 3(σ) 0.75-3(.0433) 0.75+3(0.433) 0.75-0.1299 0.75+0.1299 0.6201 0.8799 0.60 0.75

13. Sample Problems; • When you will up your gas tank you are often asked if you want to “Top it off”. The ODEC claims that 82% of drivers will agree to this “wasteful and dangerous practice” if asked by an attendant. For a particular driver let x=1 if they agree to top off and x=0 if they do not. For a random sample of 36 drivers: • 1) State the population distribution. • 2) Find the mean of the sample distribution. • 3) Find the standard Error. • 4) Within what interval is a sample proportion almost certainly going to land. 5) Your survey of 36 drivers finds that 75% of those asked “topped off”. Would this be a unusual result? Explain.

14. Lets play with an Applet It’s on your CD.