Chapter 19 Confidence intervals for proportions math2200

Sample proportion A good estimate of the population proportion Natural sampling variability How does the sample proportion vary from sample to sample? Approximately normal About 95% of all samples have within 2 sd of p Population proportion, p, is unknown Standard deviation is unknown

We can estimate! Standard error estimates standard deviation If estimates p accurately, by the 68-95-99.7% Rule, we know about 68% of all samples will have ’s within 1 SE of p about 95% of all samples will have ’s within 2 SEs of p about 99.7% of all samples will have ’s within 3 SEs of p

What can we say about p? The true value of p is . (Wrong!) The true value of p is close to . (OK!) We are 95% confident that the true value of p is between and This is a 95% confidence interval of p In this specific context, we can also call it one-proportion z-interval

Margin of error The extent of the confidence interval on either side of is called the margin of error The margin of error for our 95% confidence interval is 2se. For the 99.7% confidence interval, the margin of error is 3se. The more confident we want to be, the larger the margin of error must be. Certainty versus precision Confidence interval: estimate ± margin of error

What Does “95% Confidence” Really Mean? Each confidence interval uses a sample statistic to estimate a population parameter. But, since samples vary, the statistics we use, and thus the confidence intervals we construct, vary as well.

What Does “95% Confidence” Really Mean? (cont.) The figure to the right shows that some of our confidence intervals capture the true proportion (the green horizontal line), while others do not.

What Does “95% Confidence” Really Mean? (cont.) Our confidence is in the process of constructing the interval, not in any one interval itself. Thus, we expect 95% of all “95%-confidence intervals” to contain the true parameter that they are estimating.

Margin of Error: Certainty vs. Precision We can claim, with 95% confidence, that the interval contains the true population proportion. The extent of the interval on either side of is called the margin of error (ME). In general, confidence intervals have the form estimate ± ME. The more confident we want to be, the larger our ME needs to be.

Margin of Error: Certainty vs. Precision (cont.)

Margin of Error: Certainty vs. Precision (cont.) To be more confident, we wind up being less precise. We need more values in our confidence interval to be more certain. Because of this, every confidence interval is a balance between certainty and precision. The tension between certainty and precision is always there. Fortunately, in most cases we can be both sufficiently certain and sufficiently precise to make useful statements.

Margin of Error: Certainty vs. Precision The choice of confidence level is somewhat arbitrary, but keep in mind this tension between certainty and precision when selecting your confidence level. The most commonly chosen confidence levels are 90%, 95%, and 99% (but any percentage can be used).

Critical value The number of SE’s affects the margin of error This number is called the critical value, denoted by z* For a 95% confidence interval, the precise critical value is 1.96 For a 90% confidence interval, the precise critical value is 1.645

Critical Values (cont.) Example: For a 90% confidence interval, the critical value is 1.645:

Margin of error A confidence interval too wide is not very useful How large a margin of error can we tolerate? Reduce level of confidence makes margin of error smaller (not a good idea, though) You should think about this ahead of time, when you design your study. Choose a larger sample! Generally, a margin of error of 5% or less is acceptable

Sample size Suppose a candidate is planning a poll and wants to estimate voter support within 3% with 95% confidence. How large a sample does she need? But we do not know before we conduct the poll! However,

Sample size To be conservative, we set Solve for n, we have n = 1067.1 So, we will need at least 1068 respondents to keep the margin of error as small as 3% with a confidence level of 95% In practice, you may need more since many people do not respond If the response rate is too low, then the study might be a voluntary response study, which can be biased To cut the se in half, we must quadruple the sample size n

Assumptions Independence Plausible independence condition Randomization condition Sampled at random? From a properly randomized experiment? 10% condition Sampling without replacement can be viewed roughly the same as sampling with replacement

Assumptions Sample size assumption Normal approximation comes from the CLT Sample size must be large enough Check the success/failure condition

Example In May 2002, the Gallup Poll asked 537 randomly sampled adults “generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?” 53% answered “fairly” 7% said “don’t know” What can we conclude from this survey?

Example Plausible independence? (Yes.) Randomization condition? (Yes.) Success/failure condition (Yes.) Let’s find a 95% confidence interval Standard error: Margin of error: 95% confidence interval

A deeper understanding of confidence interval How should we interpret the 95% confidence? Randomness is from sample to sample This does not say p is random We are the one who is uncertain, not the parameter. The interval itself is random (varies from sample to sample) This interval is about p, not about The sample proportion varies Report confidence interval or margin or error

Always pay attention to the required assumptions Independence Sample size In practice, watch out for biased samples

What have we learned? Finally we have learned to use a sample to say something about the world at large. This process (statistical inference) is based on our understanding of sampling models, and will be our focus for the rest of the book. In this chapter we learned how to construct a confidence interval for a population proportion. And, we learned that interpretation of our confidence interval is key—we can’t be certain, but we can be confident.