1. 1 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Chapter 16 Confidence Intervals for Proportions

2. 2 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Objectives 57. Determine and interpret the margin of error in the context of the problem. 58. Construct a confidence interval for a proportion and interpret in the context of the problem, checking the necessary assumptions. 59. Determine the sample size necessary to produce a specific margin of error.

3. 3 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 16.1 A Confidence Interval

4. 4 Copyright © 2014, 2012, 2009 Pearson Education, Inc.

5. 5 Standard Error for a Proportion What is the sampling distribution? Usually we do not know the population proportion p. We cannot find the standard deviation of the sampling distribution: After taking a sample, we only know the sample proportion, which we use as an approximation. The standard error is given by

6. 6 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Facebook Daily Status Updates A recent survey found that 48 of 156 or 30.8% update their Facebook status daily. The sampling distribution is approximately normal.

7. 7 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Interpreting this Normal Curve By normality, about 95% of all possible samples of 156 young Facebook users will have ’s within 2 SE’s of p. If is close to p, then p is close to. If you stand at, then you can be 95% sure that p is within 2SE’s from where you are standing.

8. 8 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What You Cannot Say About p if You Know 30.8% of all Facebook users update their status daily. We can’t make such absolute statements about p. It is probably true that 30.8% of all Facebook users update their status daily. We still cannot commit to a specific value for p, only a range. We don’t know exactly what percent of all Facebook users update their status daily, but we know it is within the interval 30.8% ± 2 X 3.7%. We cannot be certain it is in this interval.

9. 9 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What You Can Say About p if You Know We don’t know exactly what percent of all Facebook users update their status daily, but the interval from 23.4% and 38.2% probably contains the true proportion. Note, we admit we are unsure about both the exact proportion and whether it is in the interval. We are 95% confident that between 23.4% and 38.2% of all Facebook users update their status daily. Notice “% confident” and an interval rather than an exact value are stated.

10. 10 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Naming the Confidence Interval This confidence interval is a one-proportion z-interval. “One” since there is a single survey question. “Proportion” since we are interested in the proportion of Facebook users who update their status daily. “z-interval” since the distribution is approximately normal.

11. 11 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 11 One-Proportion z-Interval When the conditions are met, we are ready to find the confidence interval for the population proportion, p. The confidence interval is where The critical value, z*, depends on the particular confidence level, C, that you specify.

12. 12 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Example Your local newspaper polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. p ^ = 144/330 =.4364 SE(p ^ ) = √(.4364)*(1-.4364)/330) = 0.0273 We can be 95% sure that p falls within 2 SEs of p ^, so the real p is between (.4364-2*.0273) =.38 and (.4364+2*.0273)=.49 with 95% confidence. In other words: We estimate that the support for passing the school budget is between 38% and 49% with 95% confidence. Slide 1- 12

13. 13 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Using the TI for finding a confidence interval for p: 1-PropZInt A random sample of 416 teenagers showed that 76.4% of them had experimented with alcohol. What is the 95% confidence interval for the proportion of the population? Stat…Tests…#A for 1-Prop Zint enter x: number of teenagers that have experimented with alcohol (has to be a whole number…...764*416 which is approximately 318; always go UP to the next whole number) n: 416 C-level:.95 You should get a confidence interval from 72.4% to 80.5%. We are 95% confident that the proportion of teenagers that experimented with alcohol is between 72.4 and 80.5 percent. Slide 1- 13

14. 14 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 16.2 Interpreting Confidence Intervals: What Does 95% Confidence Really Mean?

15. 15 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Capturing a Proportion The confidence interval may or may not contain the true population proportion. Consider repeating the study over an over again, each time with the same sample size. Each time we would get a different. From each, a different confidence interval could be computed. About 95% of these confidence intervals will capture the true proportion. 5% will be duds.

16. 16 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Simulating Confidence Intervals

17. 17 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Confidence Intervals There are a huge number of confidence intervals that could be drawn. In theory, all the confidence intervals could be listed. 95% will “work” (capture the true proportion). 5% will be “duds” (not capture the true proportion). What about our confidence interval (0.234, 0.382)? We will never know whether it captures the population proportion.

18. 18 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Confidence Interval on Global Warming Yale and George Mason University interviewed 1010 US adults about beliefs and attitudes on global warming. They presented a 95% confidence interval on the proportion who think there is disagreement among scientists. Had the polling been done repeatedly, 95% of all random samples would yield confidence intervals that contain the true population proportion of all US adults who believe there is disagreement among scientists.

19. 19 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Facebook Status Updates Technically Correct I am 95% confident that the interval from 23.4% to 38.2% captures the true proportion of Facebook users who update daily. More Casual But Fine I am 95% confident that between 23.4% and 38.2% of Facebook users update daily.

20. 20 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 16.3 Margin of Error: Certainty vs. Precision

21. 21 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Margin of Error Confidence interval for a population proportion: The distance,, from is called the margin of error. Confidence intervals also work for means, regression slopes, and others. In general, the confidence interval has the form

22. 22 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Critical Values For a 95% confidence interval, the margin of error was 2SE. The 2 comes from the normal curve. 95% of the area is within about 2SE from the mean. In general the number of SE is called the critical value. Since we use the normal distribution here we denote it z*.

23. 23 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Finding the Critical Value with a TI-83/84 Find the critical value corresponding to 90% confidence. 90% confidence refers to the Central 90% of the Normal curve. The z* critical values are the cut off scores. We can find this using invNorm(.05) We use.05 because (100%-90%)/2 = 5% in the negative tail We get -1.645, which we turn into a positive result z* since z* is the distance from the center The critical value is about z* = 1.645.

24. 24 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Finding the Critical Value with StatCrunch Find the critical value corresponding to 90% confidence. Stat → Calculators → Normal 90% inside gives 10% outside. 2 tails outside with 10% means 1 tail with 5% or 0.05. The critical value is about z* = 1.645.

25. 25 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Certainty vs. Precision Instead of a 95% confidence interval, any percent can be used. Increasing the confidence (e.g. 99%) increases the margin of error. Decreasing the confidence (e.g. 90%) decreases the margin of error.

26. 26 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Yale/George Mason Study Revisited The poll of 1010 adults reported a margin of error of 3%. By convention, 95% with p = 0.5. How was the 3% computed? For 95% confidence ME = 2(0.0157) = 0.031 The margin of error is close to 3%.

27. 27 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Finding the Margin of Error -- Hint If you have already solved for the confidence interval (lower, upper) using 1PropZInt on your calculator, you can find the margin of error by finding the distance from each end point to the midpoint p^. i.e. (upper - lower) / 2 (don’t forget parentheses when using your calculator!!!)

28. 28 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 16.4 Assumptions and Conditions

29. 29 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Independence and Sample Size Independence Condition If data is collected using SRS or a randomized experiment → Randomization Condition Some data values do not influence others. Check for the 10% Condition: The sample size is less than 10% of the population size. Success/Failure Condition There must be at least 10 successes. There must be at least 10 failures.

30. 30 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Do You Believe the Death Penalty is Applied Fairly? Sample size: 510 Answers: 58% “Fairly” 36% “Unfairly” 7% “Don’t Know” Construct a confidence interval for the population proportion that would reply “Fairly.”

31. 31 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Do You Believe the Death Penalty is Applied Fairly? Think → Plan: Find a 95% confidence interval for the population proportion. Model: Randomization: Randomly selected by Gallup Poll 10% Condition: Population is all Americans Success/Failure Condition (510)(0.58) = 296 > 10, (510)(0.42) = 214 > 10 Use 1-PropZInt to find a one-proportion z -interval.

32. 32 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Do You Believe the Death Penalty is Applied Fairly? Tell→ Conclusion: I am 95% confident that between 57.3% and 62.3% of all US adults think that the death penalty is applied fairly.

33. 33 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Example An experiment finds that 27% of 53 subjects report improvement after using a new medication. Create a 95% confidence interval for the actual cure rate. First check conditions: randomization condition, 10% condition, success/failure condition Why is our interval so wide? Make it narrower – 90% confidence. What are the advantages and disadvantages? Slide 1- 33

34. 34 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 8. A poll found that 38% of a random sample of 1012 American adults said that they believe in ghosts. a. Find the margin of error for this poll if we want 90% confidence in our estimate of American adults who believe in ghosts. E=0.025 b. Explain what a “90% confidence interval” means and find the interval. We can be 90% confident that the true population proportion (i.e. the percentage of American adults who believe in ghosts) is contained in the following CI: (0.355,0.405) c. If we want to be 99% confident, will the margin of error be larger or smaller? Larger d. Find that margin of error. E=0.0393 e. In general, will smaller margins of error involve greater or less confidence in the interval? Less Slide 1- 34

35. 35 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 9. Direct mail advertisers send solicitations to thousands of potential customers in the hope that some will buy the company’s product. The response rate usually is quite low. Suppose a company wants to test the response to a new flyer and sends it to 1000 people randomly selected from their mailing list of over 200,000 people. They get 123 orders from the recipients. a. Create a 90% confidence interval for the percentage of people the company contacts who may buy something. (10.6%, 14.0%) b. Explain what the interval means. We can be 90% confident that the true proportion of the company’s 200,000 customers who will actually purchase an item is in the above interval. c. The company must decide whether to now do a mass mailing. The mailing won’t be cost effective unless it produces at least a 5% return. What does your confidence interval suggest? They should do the mass mailing. Slide 1- 35

36. 36 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 10. A national health organization warns that 30% of the middle school students nationwide have been drunk. Concerned, a local health agency randomly and anonymously surveys 110 of the 1212 middle school students in its city. Only 21 of them reported having been drunk. a. What proportion of the sample reported having been drunk? 21/110 = 0.191 b. Does this mean that this city’s youth are not drinking as much as the national data would indicate? Not necessarily – we need to build a confidence interval. c. Create a 95% confidence interval for the proportion of the city’s middle school students who have been drunk. (11.7%, 26.4%) d. Is there any reason to believe that the national level of 30% is not true of the middle school students in this city? Yes – even the upper bound of the above confidence interval is below 30%. Slide 1- 36

37. 37 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Sample Size Determination for Estimating a Population Proportion The margin of error is z*SE(p ^ )… solving for n we get: where: z * = critical value based on the desired confidence level E = desired margin of error If p ^ and q ^ are unknown, assume they are both.5, this is the “worst case scenario”

38. 38 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Choosing a sample size An experiment finds that 27% of 53 subjects report improvement after using a new medication. What sample size would we need in a follow up study if we want a margin of error ± 5% with 98% confidence? Solve for z* of the 98% confidence interval using invNorm(.01) = -2.33 Recall the formula: Set E =.05, z=2.33, use our previous estimates of p ^ and q ^ and solve for n. Hint: always round up because we need to ensure a large enough sample. Slide 1- 38

39. 39 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Example on choosing a sample size 11. In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days. a. How many randomly selected employers must we contact in order to create an estimate in which we are 95% confident with a margin of error of 5%? 385 b. Suppose we want to reduce the margin of error to 3%. What sample size will suffice? 1068 c. Why might it not be worth the effort to try to get an interval with a margin of error of only 1%? Because it would take a sample size of 9,604. That is a mighty big (i.e. expensive) number. Slide 1- 39

40. 40 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Thoughts on Sample Size and ME Obtaining a large sample size can be expensive and/or take a long time. For a pilot study, ME = 10% can be acceptable. For full studies, ME < 5% is better. Public opinion polls typically use ME = 3%, n = 1000. If p is expected to be very small such as 0.005, then much smaller ME such as 0.1% is required.

41. 41 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Credit Cards and Sample Size A pilot study showed that 0.5% of credit card offers in the mail end up with the person signing up. To be within 0.1% of the true rate with 95% confidence, how big does the test mailing have to be? n ≈ 19,111.96 The test mailing should include at least 19,112 offers.

42. 42 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What Can Go Wrong? Don’t claim other samples will agree with yours. Wrong: In 95% of samples, between 43% and 51% agree with decriminalization of marijuana. Don’t be certain about the parameter. Wrong: Between 23% and 38% of Facebook users update daily. Don’t forget to include the confidence. Don’t forget that it’s about the parameter. Wrong: I’m 95% confident that is between 23% and 38%. You know for sure exactly what is.

43. 43 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What Can Go Wrong? Don’t claim to know too much. Wrong: I’m 95% confident that between 23% and 38% of all Facebook users in the world update daily. The survey was just about US residents between 18 and 22. Do take responsibility. Accept that you are only 95% confident, not sure. Don’t suggest that the parameter varies. Wrong: There is a 95% chance that the true parameter is between 23% and 38%.

44. 44 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What Can Go Wrong? Do treat the whole interval equally. The middle of the interval is not necessarily more plausible than the edges. Beware of margins of error that are too large to be useful. Between 10% and 90% update daily is not useful. Use a larger sample size to shrink the ME. Watch out for biased sampling. Biased samples produce an unreliable CIs. Think about independence Be careful in your sample design to ensure randomization.