Hypothesis Tests with Proportions

1. Hypothesis Tests with Proportions

2. What are hypothesis tests? These calculations (called the test statistic) will tell us how many standard deviations a sample proportion is from the population proportion! Is it one of the sample proportions that are likely to occur? 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? Statistically significant means that it is NOT a random chance occurrence! Is it one that isn’t likely to occur?

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

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

5. Assumptions for z-test: Have an SRSof context Distribution is (approximately) normalbecausebothnp > 10andn(1-p) > 10 Population is at least 10n YEA – These are the same assumptions as confidence intervals!!

6. Check assumptions for the following: • Given SRS of homes • 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. 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.

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: Ha:

8. How to write hypotheses: Null hypothesis H0: parameter = hypothesized value Alternative hypothesis Ha: parameter > hypothesized value Ha: parameter < hypothesized value Ha: 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? H0: p = .7 Ha: p < .7 Where p is the true proportion of vaccinated people who do not get the flu

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 : H0: m = 40 Ha: m = 40 Where m is the true mean amperage of the fuses

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. Must use parameter (population) x is a statistics (sample) Activity: For each pair of hypotheses, indicate which are not legitimate & explain why Must be NOT equal! p is the population proportion! Must use same number as H0! P-hat is a statistic – Not a parameter!

13. P-values - • Assuming H0 is true,the probability that the statistic would have a value as extreme or morethan 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 a • Can be any value • Usual values: 0.1, 0.05, 0.01 • Most common is 0.05

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

17. Facts about p-values: • ALWAYSmake 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 accept the null hypothesis!

18. Never“accept” the null hypothesis! Never“accept” the null hypothesis! Never“accept” the null hypothesis!

19. At an alevel of .05, would you reject or fail to reject H0 for the given p-values? • .03 • .15 • .45 • .023 Reject Fail to reject Fail to reject Reject

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

21. Draw & shade a curve & calculate the p-value: • right-tail test z = 1.6 2) two-tail test z = -2.4 P-value = .0548 P-value = .0164

22. Writing Conclusions: • A statement of the decision being made (reject or fail to reject H0) & why (linkage) • A statement of the results in context. (state in terms of Ha) AND

23. “Since the p-value < (>) a, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha.” Be sure to write Ha in context (words)!

24. H0: p = .7 Ha: p < .7 Where p is the true proportion of vaccinated people who get the flu P-value = normalcdf(-10^99,-1.38) =.0838 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 a = 0.05. Since the p-value > a, I fail to reject H0. 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. Example 5: A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad and recognize the product. Is this sufficient evidence for the company to renew its contract?

27. Assumptions: • Have an SRS of people • np = 400(.2) = 80 & n(1-p) = 400(.8) = 320 - Since both are greater than 10, this distribution is approximately normal. • Population of people is at least 4000. Use the parameter in the null hypothesis to check assumptions! H0: p = .2 where p is the true proportion of people who Ha: p > .2 heard the ad Use the parameter in the null hypothesis to calculate standard deviation! Since the p-value > a, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true proportion of people who heard the ad is greater than .2. The company will not renew their advertising contract with the radio station.

28. Calculate the appropriate confidence interval for the above problem. CI = (.19066,.25934) How do the results from the confidence interval compare to the results of the hypothesis test? The confidence interval contains the parameter of .2 thus providing no evidence that more than 20% had heard the ad.