Hypothesis Tests with Proportions

1. Hypothesis Tests with Proportions Chapter 10

2. Write down the first number that you think of for the following . . . Pick a two-digit number between 10 and 50, where both digits are ODD and the digits do not repeat.

3. What possiblevalues fit this description? • Record your answer on the dotplot on the board. • What do you notice about this distribution? • Did you expect this to happen?

4. What proportion of the time would I expect to get the value 37 if the values were equally likely to occur? • Is the difference in these proportions significant? A hypothesis test will help me decide! How do I know if this p-hat is significantly different from the 1/8 that I expect to happen?

5. 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?

6. 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!

7. 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

8. 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!!

9. 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.

10. 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:

11. How to write hypotheses: Null hypothesis H0: parameter = hypothesized value Alternative hypothesis Ha: parameter > hypothesized value Ha: parameter < hypothesized value Ha: parameter = hypothesized value

12. Example 2:(Back to the opening activity) Is the proportion of students who answered 37 higher than the expected proportion of 1/8? H0: p = 1/8 Ha: p > 1/8 Where p is the true proportion of people who answered “37”

13. 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

14. 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

15. 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!

16. 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!

17. Level of Significance Activity

18. P-value - The statistic is our p-hat! • Assuming H0 is true,the probability that the statistic would have a value as extreme or morethan what is actually observed Notice that this is a conditional probability Why not find the probability that the p-hat equals a certain value? Remember that in continuous distributions, we cannot find probabilities of a single value!

19. P-values - We can use normalcdf to find this probability. • Assuming H0 is true,the probability that the statistic would have a value as extreme or morethan what is actually observed In other words . . . What is the probability of getting values more (or less) than our p-hat?

20. 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

21. Statistically significant – Remember that the verdict is never “innocent” – so we can never decide that the null is true! • Our statistic (p-hat) is statistically significant if the p-value is as small or smaller than the level of significance (a). Decisions: • If p-value <a, “reject” the null hypothesis at the a level. • If p-value > a, “fail to reject” the null hypothesis at the a level. Our “guilty” verdict. Our “notguilty” verdict.

22. Facts about p-values: • ALWAYSmake the 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!

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

24. Calculating p-values • For z-test statistic (z) – • Use normalcdf(lb,ub) to find the probability of the test statistic or more extreme • Remember the standard normal curve is comprised of z’s where m = 0 and s = 1 We will see how to compute this value tomorrow. Since we are in the standard normal curve, we do not need m, s here.

25. Draw & shade a curve & calculate the p-value: • right-tail test z = 1.6 2) two-tail test z = -2.4 Normalcdf(1.6,∞) P-value = .0548 Double the p-value since this is a two-tailed test! z Normalcdf(-∞,-2.4) × 2 P-value = .0164 z

26. 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

27. 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

28. “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)!

29. 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%.

30. Formula for hypothesis test:

31. Let’s put all the steps together! Example 2 revisited: Is the proportion of people who think of the value 37 significantly higher than what we expect? Use a = 0.05.

32. What confidence level would be equivalent to this right-tailed test with a = 0.05? Calculate this confidence interval. How do the results from the confidence interval compare to the results of the hypothesis test?

33. 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?

34. 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.

35. Calculate the appropriate confidence interval for the above problem. = .225 + .041 = (.184, .266) 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.