Inferences about Single Sample Proportions Confidence Intervals

Confidence Intervals: An Analogy The general form of a 95% confidence interval is as follows: sample statistic ± 2 x (standard error)

Example of possible confidence intervals

Estimating Proportions with Confidence Thought Questions A 95% confidence interval for the proportion of adults in the U.S. who have diabetes extends from .07 to .11, or 7% to 11%. What does it mean to say that the interval from .07 to .11 represents a 95% confidence interval for the proportion of adults in the U.S. who have diabetes ? 2. Do you think a 99% confidence interval for the proportion described in Question 1 would be wider or narrower than the 95% interval given? Explain.

Estimating Proportions with Confidence Red dot represents the sample proportion for each sample Green horizontal line represents the true population proportion. Blue lines represent the width of the confidence interval

Assumptions and Conditions Here are the assumptions and the corresponding conditions you must check before creating a confidence interval for a proportion: Independence Assumption: We first need to Think about whether the Independence Assumption is plausible. It’s not one you can check by looking at the data. Instead, we check two conditions to decide whether independence is reasonable.

Assumptions and Conditions Randomization Condition: Were the data sampled at random or generated from a properly randomized experiment? Proper randomization can help ensure independence. 10% Condition: Is the sample size no more than 10% of the population? Sample Size Assumption: The sample needs to be large enough for us to be able to use the CLT. Success/Failure Condition: We must expect at least 10 “successes” and at least 10 “failures.”

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 that you specify.

Example: Binge drinking among College Students Example: If the random sample of 2166 college graduates in California found that 279 had engaged in binge drinking in 2006, the sample proportion is 0.129. Construct a confidence interval for the proportion of college students in California engaged in binge drinking in 2006. A 95% confidence interval for a population proportion: sample proportion ± 1.96(SE) where

Example Let’s say that we are interested in estimating the population proportion of people who are going to vote for a particular candidate in an election. We select a random sample of 500 voters and ask them who they are going to vote for. We find that 200 out of the 500 voters plan to vote for the candidate, a sample proportion of 200/500 equal to 0.40. Our 95% confidence interval for the population proportion is: 0.4 ± 2 x (0.02) 0.4 ± 0.04 [0.36,0.44]

Example: Binge drinking among College Students The 95% confidence interval for the proportion college graduates in California who engaged in binge drinking in 2006 is 0.129 ± 1.96 (0.129)(1 – 0.129) 2166 = 0.129 ± (1.96) (0.0072) = 0.129 ± 0.014 = 0.115 to 0.143 We are 95% confident that the true population proportion (percentage) of all college graduates in California who engaged in binge drinking in 2006 lies between 11.5% and 14.3%.

Example : Experiment in ESP Experiment: Subject tried to guess which of four videos the “sender” was watching in another room. Of the 165 cases, 61 resulted in successful guesses. sample proportion = 61/165 = 0.37 or 37% standard error of sample proportion: 95% confidence interval = .37 ± 1.96(0.038) = .37 ± .075 = [0.295 to 0.445] or 29.5% to 44.5% (0.37)(1 – 0.37) = 0.038 165

Understanding the Confidence Interval If we collect more data, reduce standard error, what would happen to the width of confidence interval? A. Wider B. Narrower C. No change

Understanding the Confidence Level For a confidence level of 95%, we expect that about 95% of all such intervals will actually cover the true population value. The remaining 5% will not. Confidence is in the procedure over the long run. 90% confidence level => multiplier = 1.645 95% confidence level => multiplier = 1.96 99% confidence level => multiplier = 2.576 More confidence  Wider Interval (for the standard error)

Political Polarization & Media Habits When it comes to getting news about politics and government, liberals and conservatives inhabit different worlds. There is little overlap in the news sources they turn to and trust. And whether discussing politics online or with friends, they are more likely than others to interact with like-minded individuals, according to a new Pew Research Center study. http://www.journalism.org/2014/10/21/political- polarization-media-habits/  

Political Polarization & Media Habits One of the main findings was that 47% of Americans whom the report defines as “consistently conservative” name Fox News as their main source for news on politics. The sample size was equal to 309 consistent conservatives with a margin of error of 7.2 percent (or 0.072).    margin of error = 2 x standard error

Political Polarization & Media Habits For the confidence interval to be valid, we have to assume the individual responses are independent from each other, and that the sample size is sufficiently large. The sample size of 309 consistent conservatives is large enough to satisfy the sample size condition for proportions.   The 95% confidence interval is calculated as follows: 0.47 ± 0.072 [0.398,0.542]

Text Question 25 Teenage drivers An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them. a) Create a 95% confidence interval for the percentage of all auto accidents that involve teenage drivers. b) Explain what your interval means. c) Explain what “95% confidence” means. d) A politician urging tighter restrictions on drivers’ licenses issued to teens says, “In one of every five auto accidents, a teenager is behind the wheel.” Does your confidence interval support or contradict this statement? Explain.

What Can Go Wrong? Interpret your confidence interval correctly. Many statements that sound tempting are, in fact, misinterpretations of a confidence interval for a mean. A confidence interval is about the mean of the population, not about the means of samples, individuals in samples, or individuals in the population.

What Can Go Wrong? Don’t Misstate What the Interval Means: Don’t suggest that the parameter varies. Don’t claim that other samples will agree with yours. Don’t be certain about the parameter. Don’t forget: It’s about the parameter (not the statistic). Don’t claim to know too much. Do take responsibility (for the uncertainty). Do treat the whole interval equally.