1. Final Project Some details on your project –Goal is to collect some numerical data pertinent to some question and analyze it using one of the statistical tests we’ve discussed in class –You will be graded on all aspects of the task from the nature of the question to the execution of the statistical test

2. Final Project Some examples: –Does the price of oil correlate with the price of gasoline? Approach: record daily price of oil and the price of gas at some gas station over several weeks and run a correlation –Is Calgary colder/windier/rainer than Edmonton Collect data from Environment Canada’s web site –Do Canadians score more than other NHL players? Collect data from any sports section or website

3. Final Project Guidelines: –Use readily available observational data Don’t run an experiment unless you check with me first!!! –Keep questions simple and straightforward Get your idea checked by Farshad before you proceed –Plan to do your project with Excel or some stats program Turn in the data, the relevant statistics, and one or two sentences explaining your question and the answer - should fit on one page.

4. Some Review A population is a really big bunch of numbers

5. Some Review A population is a really big bunch of numbers A sample is some of the numbers from a population

6. Some Review All sets of numbers have a distribution –The population has a mean –A sample has a mean that is probably similar but not necessarily the same as the population

7. Some Review All sets of numbers have a distribution –The population has a standard deviation –A sample has a standard deviation that is probably similar but not necessarily the same as the population

8. Some Review If we think in terms of standard deviation, we can know things like whether or not a single number is very different from the mean of a population

9. Some Review But often we’re not interested in single numbers - we’ve collected a sample and computed a mean That mean comes from a population of sample means (you just happened to pick one of them) The mean of the distribution of sample means is the mean of the population The standard deviation of the sample means is the standard error

10. Some Review If we think in terms of standard errors, we can know things like whether a particular mean is very different from the mean of a population

11. Keep these ideas straight If we think in terms of standard deviation, we can know things like whether or not a single number is very different from the mean of a distribution If we think in terms of standard errors, we can know things like whether a particular mean is very different from the mean of a population

12. Some Review We use the Z table to look up the probability that a particular Z score came from any normal population

13. Some Review We use the Z table to look up the probability that a particular Z score came from any normal population Since the population of sample means is normal (Central Limit Theorem), we can use the same Z table to look up the probability that a sample mean came from a population with a particular mean

14. Now a Real Example Break into groups of 10 Write down your heights in inches Compute the mean of your n=10 sample Compute the standard deviation Hand it all in to Fraser

15. Critical Z Value In our examples we’ve been testing the hypothesis that one sample has a mean that is higher (or lower) than a population mean

16. Critical Z Value In our examples we’ve been testing the hypothesis that one sample has a mean that is higher (or lower) than a population mean Let’s turn this around a bit…let’s work backwards

17. Critical Z Value How much bigger would a sample mean have to be so that there’s only a 5% chance that it came from a particular population?

18. Critical Z Value How much bigger would a sample mean have to be so that there’s only a 5% chance that it came from a particular population? 95% This is the alpha =.05 threshold What Z score? 5%

19. Critical Z Value This is sometimes called the critical Z value or

20. Directional vs. Bidirectional Tests In our examples we’ve been testing the hypothesis that one sample has a mean that is higher (or lower) than a population mean We call this a directional or “one-tailed” test What does that one-tailed bit mean !?

21. Directional vs. Bidirectional Tests We were checking to see if our sample had a mean far enough into the positive tail of the distribution and ignoring the negative tail

22. Directional vs. Bidirectional Tests Often we haven’t made a directional hypothesis, but have simply predicted “a difference”

23. Directional vs. Bidirectional Tests Often we haven’t made a directional hypothesis, but have simply predicted “a difference” In that situation, we are twice as likely to make a Type I error: the sample mean could, by chance, be in either tail !

24. Directional vs. Bidirectional Tests What would the critical Z value be so that there is a 5% chance that a mean is beyond it in either direction?

25. Directional vs. Bidirectional Tests What would the critical Z value be so that there is a 5% chance that a mean is beyond it in either direction? 95% This is the alpha =.05 threshold What Z score? 2.5%

26. Directional vs. Bidirectional Tests Thus: