Significance Testing

1. Significance Testing Statistical testing of the mean (z test)

2. Binomial Distribution • Mathematicians have figured formulas to estimate long run relative frequencies for simple events, like how many heads will appear for a given number of coin tosses. The binomial is one such. Recall dice Number of ‘heads’ in 10 flips of a coin.

3. We have already figured percentages of the normal. Percentages of the normal correspond to probabilities of finding individual cases in the distribution. The sampling distribution of the mean is normal if N is large. Normal Distribution Middle 95 percent from going up and down 1.96 SDs from the mean.

4. Significance testing is a ‘what if’ game. We make an assumption, and ask what will happen if the assumption is true. Assumption is null hypothesis. Significance testing is based on probabilities that come from the ‘what if’ scenario (from the null hypothesis). What if the true mean height of students at USF is 66 inches and SD is 5 inches? What if we draw people from USF 100 at a time and plot the means? We can figure the sampling distribution from these assumptions and figure probabilities. Rejection region is place that is unlikely to occur if null is true. Significance Testing 1

5. Significance Testing 2 • Given: • Result 1: Sampling distribution of mean is normal (mu =66). • Result 2: Standard error of the mean is: Rejection Region Rejection Region What is a rejection region?

6. Review • Suppose population mean is 500, population SD is 100 (SAT data), and sample size is 100. • Draw sampling distribution of means. • What is the shape of this distribution? • What is the mean of this distribution? • What is the standard deviation of this distribution? • Find, mark, and label the rejection regions.

7. Review Upper = 500+1.96(10)=519.6 Lower = 500-1.96(10) =480.4 RR > 519.6 RR < 480.4 RR RR Shape is normal, mean is 500, SD is 10.

8. Significance Testing 3 • Establish ‘what if’ • Collect sample data. • Examine probability of sample result given the null. • If probability is low, say that null is false, i.e., reject the null hypothesis. • This is a significance test. If we reject the null, we say result is statistically significant. • Significance testing lets us make decisions about populations from sample data.

9. Example • Mean # beers at Skipper’s Smokehouse? • Null (what if): • Data from Skipper’s: Derive: Reject the null. Result is significant. Observed data are very unlikely if null is true. Null must be false. Lots of beer at Skipper’s. Note: data are fictitious.

10. Review • We want to know if a workbook helps with learning stats. We know from past classes that students average 75 percent on the final with a SD of 5. Our new class of 225 has a mean of 78. Did the workbook help? (Hint: sqrt(225) = 15.)

11. Review Upper = 75+1.96(.33) = 75.65. This is far below 78. The workbook helps.

12. Definition • The term probability refers to the long run • 1 Frequency of outcome • 2 Odds ratio • 3 Relative frequency • 4 Rolling of dice

13. Definition • The calculation of probabilities in hypothesis testing rests upon assumptions described in the ______. • 1 Alternative hypothesis • 2 Null hypothesis • 3 Sampling distribution • 4 Standard error

14. Definition • The rejection region is the place in the sampling distribution that is _____ • Close to the mean • Obtained when the result is not significant • Very unlikely if the null hypothesis is true • Visited by losers