1. Hypothesis Tests with Proportions Chapter 10 Notes: Page 169

2. What are hypothesis tests? Calculations that tell us if the sample statistics (p-hat) occurs by random chance or not OR... if it is statistically significant Is it... –a random occurrence due to natural variation? –an occurrence due to some other reason? NOT Statistically significant means that it is NOT a random chance occurrence! Is it one of the sample proportions that are likely to occur? Is it one that isn ’ t likely to occur? test statistic These calculations (called the test statistic) will tell us how many standard deviations a sample proportion is from the population proportion!

3. Nature of hypothesis tests - First begin by supposing the “ effect ” is NOT present Next, see if data provides evidence against the supposition Example:murder trial How does a murder trial work? First - assume that the person is innocent must Then – must have sufficient evidence to prove guilty Hmmmmm … Hypothesis tests use the same process!

4. Steps: 1)Assumptions 2)Hypothesis statements & define parameters 3)Calculations 4)Conclusion, in context Notice the steps are the same as a confidence interval except we add hypothesis statements – which you will learn today

5. Assumptions for z-test: Have an SRS of context Distribution is (approximately) normal because both np > 10 and n(1-p) > 10 Population is at least 10n YEA YEA – These are the same assumptions as confidence intervals!!

6. Check assumptions for the following: Example 1: A countywide water conservation campaign was conducted in a particular county. A month later, a random sample of 500 homes was selected and water usage was recorded for each home. The county supervisors wanted to know whether their data supported the claim that fewer than 30% of the households in the county reduced water consumption after the conservation campaign. Given SRS of homesGiven SRS of homes Distribution is approximately normal because np=150 & n(1-p)=350 (both are greater than 10)Distribution is approximately normal because np=150 & n(1-p)=350 (both are greater than 10) There are at least 5000 homes in the county.There are at least 5000 homes in the county.

7. How to write hypothesis statements Null hypothesis – is the statement (claim) being tested; this is a statement of “ no effect ” or “ no difference ” Alternative hypothesis – is the statement that we suspect is true H0:H0:H0:H0: Ha:Ha:Ha:Ha:

8. How to write hypotheses: Null hypothesis H 0 : parameter = hypothesized value Alternative hypothesis H a : parameter > hypothesized value H a : parameter < hypothesized value H a : parameter = hypothesized value

9. Example 3: A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. Is this claim too high? Where p is the true proportion of vaccinated people who do not get the flu H 0 : p =.7 H a : p <.7

10. Example 4: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses : Where  is the true mean amperage of the fuses H 0 :  = 40 H a :  = 40

11. Facts to remember about hypotheses: Hypotheses ALWAYS refer to populations (use parameters – never statistics) The alternative hypothesis should be what you are trying to prove! ALWAYS define your parameter in context!

12. Activity: For each pair of hypotheses, indicate which are not legitimate & explain why Must use parameter (population) x is a statistics (sample)  is the population proportion! Must use same number as H 0 ! P-hat is a statistic – Not a parameter! Must NOT be equal!

13. P-values - as extreme or moreAssuming H 0 is true, the probability that the statistic would have a value as extreme or more than what is actually observed In other words... is it far out in the tails of the distribution?

14. Level of Significance Activity

15. Level of significance - Is the amount of evidence necessary before we begin to doubt that the null hypothesis is true Is the probability that we will reject the null hypothesis, assuming that it is true Denoted by  –Can be any value –Usual values: 0.1, 0.05, 0.01 –Most common is 0.05

16. Statistically significant – as smallsmallerThe p-value is as small or smaller than the level of significance (  ) fail to rejectIf p-value > , “ fail to reject ” the null hypothesis at the  level. rejectIf p-value < , “ reject ” the null hypothesis at the  level.

17. Facts about p-values: ALWAYS make decision about the null hypothesis! Large p-values show support for the null hypothesis, but never that it is true! Small p-values show support that the null is not true. Double the p-value for two-tail (=) tests Never acceptNever accept the null hypothesis!

18. Never “ accept ” the null hypothesis!

19. At an  level of.05, would you reject or fail to reject H 0 for the given p-values? a).03 b).15 c).45 d).023 Reject Fail to reject

20. Calculating p-values For z-test statistic – –Use normalcdf(lb,ub) –Remember that z ’ s form the standard normal curve with  = 0 and  = 1

21. Draw & shade a curve & calculate the p-value: 1)right-tail test z = 1.6 2) two-tail testz = -2.4 P-value = 1-.9452 =.0548 P-value =.0082 + (1-.9918) =.0082 +.0082 =.0164 -2.4 +2.4

22. Writing Conclusions: 1)A statement of the decision being made (reject or fail to reject H 0 ) & why (linkage) 2)A statement of the results in context. (state in terms of H a ) AND

23. “ Since the p-value ) , I reject (fail to reject) the H 0. There is (is not) sufficient evidence to suggest that H a. ” Be sure to write H a in context (words)!

24. Example 3 revisited: A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. In a test, vaccinated people were exposed to the flu. The test statistic for the results is z = - 1.38. Is this claim too high? Write the hypotheses, calculate the p-value & write the appropriate conclusion for  = 0.05. H 0 : p =.7 H a : p <.7 Where p is the true proportion of vaccinated people who get the flu P-value = normalcdf(-10^99,-1.38) =.0838 Since the p-value > , I fail to reject H 0. There is not sufficient evidence to suggest that the proportion of vaccinated people who do not get the flu is less than 70%.

25. Formula for hypothesis test:

26. Homework: Page 172 and 173