1. Choosing Sample Size and Using Your Calculator Presentation 9.3

2. Margin of Error The margin of error (m) of a confidence interval is the plus and minus part of the confidence interval A confidence interval that has a margin of error of plus or minus 3 percentage points means that the margin of error m=.03. Margin of Error

3. A common problem in statistics is to figure out what sample size will be needed to obtain a desired accuracy or margin of error. This is essentially algebra problem.

4. Determining Sample Size Set up the following to obtain a margin of error m. p* is you best guess of the proportion (remember you determine sample size before you actually take the sample). –More on this p* later. Then, solve for n. Be sure to ALWAYS round up. –If you round, for example 5.023 to 5, your margin of error will come out just a hair to big. –So, err on the side of caution and ALWAYS round up!

5. Sample Size The margin of error desired m, is usually provided in the problem. The value z* is determined by the level of confidence that is desired (typically 90%, 95%, or 99%). The p* value is your best guess about the value of the true p. –So we are trying to do a study to estimate p, but we need to know p or p* to compute the needed sample size. This seems impossible! –What to do, what to do?

6. Sample Size Do the best you can. Give the best or most current state of knowledge about p as p*. Many times there is some information or hint about what p might be. If you know absolutely nothing, then use p*=.5 as that will create the largest standard error and thus guarantee your margin of error. –This is again erring on the side of caution.

7. Why use p*=.5? Here is a graph of p*(1-p*) for values of p*: p* p*=0.5 1 p*(1-p*).25 So you can see that using p*=.5 gives you the largest standard error.

8. Why use p*=.5 The graph shows that p*(1-p*) will be largest when p*=.5. –This means the sample size will be largest when p*=.5. –Which means that the sample size will be at least as big as actually needed. This is being conservative as you are using more data than you would actually need to achieve the desired margin of error.

9. Sample Size Example #1: Home Court Advantage Home Court Advantage In watching n=20 college basketball games, it seems as if the home team usually wins. In fact, the home team won 14 times in 20 games. This means p-hat = 14/20 =.7 or 70% of the time! What is a 95% confidence interval for true home court win proportion p?

10. Sample Size Example #1: Home Court Advantage Calculate the confidence interval A 20% margin of error! That is unacceptable and a rather useless confidence interval! –It’s simply way too wide!

11. Sample Size Example #1: Home Court Advantage How big of a sample would we need? How accurate (narrow interval or small margin of error) would we like to be? Suppose we wish to obtain a margin of error of 3% in a 95% CI for p. –That is, we want a proportion plus or minus 3%. How many games would I have to or get to watch?

12. Sample Size Example #1: Home Court Advantage Set up the equation –We need to guess p* –To be conservative, use.5 Solve for n Round up! Divide both sides by 1.96 Square both sides Multiply both sides by n Divide both sides by.000234

13. Sample Size Example #1: Home Court Advantage Very cool! I now have a statistical reason for watching 1069 college basketball games!

14. Choosing Sample Size and Using Your Calculator This concludes this presentation.