Hypothesis Testing For Means!

1. Hypothesis Testing For Means! The 10 Step Quest Continues!

2. Overview • When we hypothesis test, we are trying to investigate whether or not a sample provides strong evidence of something. • Previously, we wanted to see if there was evidence that the proportion had increased, decreased, or changed. • Now, we will do the same thing for means.

3. Step 1 – Test Selection • When we do a hypothesis test, we have to determine which test to use. • When we only have 1 sample, we use a 1 sample test. • When we have proportional data, we use a proportion test. • When we have quantitative data, we use a t test (unless we magically know ).

4. Step 1 – Test Selection • This leads to the following tests: • 1 proportion z • 1 sample z (when we magically know sigma) • 1 sample t • Each one of these tests corresponds with a confidence interval, meaning we know 3 so far.

5. Step 2 – Check Conditions • The type of test we end up doing will have 3 to 5 conditions for us to check. • For the 1 proportion z test, there are 4 conditions. • They are Random, Independent, Less than 10%, and Success/Failure. • For the 1 sample z and the 1 sample t, there are 3 conditions. • They are Random, Less than 10%, and Nearly Normal.

6. Step 2 – Check Conditions • For Random, we usually will just say that the problem says it was random. • Otherwise, we either need to give a sensible reason to presume it is random (like self-respecting scientists should use randomization) or give a sensible reason to presume it is not random.

7. Step 2 – Check Conditions • For Less Than 10%, you multiply the sample size by 10, and then state that the population has more than that amount of whatevers. • For Nearly Normal, we simply state that the absence of skew or outliers is reasonable if we do not know the data. • If we do know the data, then we look for obvious skew or outliers. • If the sample size is at least 30, we can actually ignore anything but the most ridiculous skew, although outliers are still a problem.

8. Step 3 – Write The Hypotheses. • You should start by writing the alternative hypothesis first. • The wording of the actual question is meant to provide the necessary clues on how the alternative should be written. • Writing the alternative requires making three distinct decisions. • You need to determine which variable, inequality sign, and number to use.

9. Step 3 – Write The Hypotheses • If the data is proportional, the variable is p. • If the data is quantitative (which is next unit), the variable is μ. • The wording of the question helps us determine the inequality sign. • If they are asking about finding evidence of a “change” or a “difference” then you use ≠. • If they are asking about finding evidence of an “increase” or some other such word, then you use >. • A word like “decrease” is a key word for <.

10. Step 3 – Write The Hypotheses • The number in your alternative hypotheses is not the number generated by your sample. • It will be a different number, and will usually be based on either previous data or else the known value for a similar group. • Occasionally, you will need to determine the number using critical thinking.

11. Step 4 – Write The Formula • 1 proportion z: • 1 sample z: • 1 sample t:

12. Step 5 – The Calculating, Part 1 • p and μ come from the H0. • p-hat and x-bar come from the sample. • The square root of n business can be tricky in the calculator. • You need to put your denominator in parentheses, or else the square root of n gets handled incorrectly. • Seriously. • It is a really common mistake.

13. Step 5 – The Picture • On your picture, you should label the center with p or μ, and mark the p-hat or x-bar, shading towards the tail of the graph. • If it is a ≠ test, shade both sides of the graph. • You do not need to label the other side.

14. Step 6 – The Calculating, Part 2 • Since we only want the probability from the tail, we use normalcdf, when we have z,in the following way: • If z is positive: normalcdf(z,5) • If z is negative: normalcdf(-5,z) • If we have t, we use tcdf in the following way: • If t is positive: tcdf(t,100,df) • If t is negative: tcdf(-100,t,df) • df = n – 1 • If it is a ≠ test, we double the value we get, since we need the area for both tails of the graph.

15. Step 7 – Determine Alpha • It is .05 unless the problem says otherwise. • Here are some examples of how that would happen: • Perform a hypothesis test. (α = .01) • Perform a hypothesis test at the 10% significance level. • Perform a hypothesis test using a .005 significance level. • Perform a hypothesis test using 2% significance.

16. Step 8 – Write The Decision Rules • If p-value < .05, reject H0. • If p-value ≥ .05, DNR H0. • If you have a different alpha, write that instead of .05 in the above rules.

17. Step 9 – Make Your Decision • Low p-value: “Since p-value = _____ < .05, reject H0.” • High p-value: “Since p-value = _____ > .05, DNR H0.”

18. Step 10 – Write Your Conclusion • Rejecting is evidence!!!!! • Reject H0: “There was sufficient evidence that blahblahblah.” • DNR H0: “There was not sufficient evidence that blahblahblah.” • The blahblahblah involves regurgitating the problem wording.

19. In Summary • The only real changes from our previous hypothesis testing stuff are: • We have somewhat different conditions to check. • We use μ instead of p for hypotheses and the picture. • We use x-bar instead of p-hat for the picture. • There is a different formula. • We might use tcdf and need df for the calculations.

20. Calculating By Hand Is For Chumps • We will now learn how to do the sucky part (steps 4-6) on the calculator. • What you should do is write the formula, draw the picture, and then use the calculator to get the p-value.

21. Assignments • Chapter 23: 25,33 • I will hand out the test review tomorrow. • On Tuesday, I will go over the test review. • On Wednesday, we will have a quiz. • On Thursday, we will have the test. • This gives a few extra lunches (and time on Wednesday when people are done with the quiz) for extra review. • Also, you have the weekend.