Confidence Intervals for Proportions

Standard Error When working with sample proportions, we assume the proportions ( 𝑝 ) from our sample is equal to the proportion from the population. However, we can’t actually know p, thus we need to change the way we talk about standard deviation, and instead refer to it as standard error.

Standard Error Formula 𝑆𝐸 𝑝 = 𝑝 𝑞 𝑛 Remember that 𝑝 is the sample proportion. When a true proportion cannot be known, this how we will calculate and refer to the spread of sample proportions.

Confidence! As you have noticed by now, Statistics is all about the grey area, and not making general claims. Refer to our Cereal Frappy from yesterday, you all felt uncomfortable deciding if the students had a legitimate claim against the company. Instead we want to talk about how confident we can be our decisions, essentially we want to avoid liability.

Confidence Interval What might have been easier to say yesterday would have been, “We are 95% confident that between .14 and .26 of the cereal boxes will contain a voucher. This type of statement is called a confidence interval. The example from yesterday is called a one-proportion z-interval.

What does 95% confidence mean? When we say we are “95% confident,” that means that 95% of the sample we would take would have the true proportion within our interval. Our confidence interval has the form 𝑝 ±2𝑆𝐸( 𝑝 ) The extent of the interval on either side of 𝑝 is called the margin of error (ME). The larger the ME, the more certain but less precise we are. The smaller the ME the more precise we are, but less certain.

Critical Values So far we have been using 2SE to give us a 95% confidence interval. The number of SE’s we use is our critical value. Since we are using the normal model we denote this with z*. For a 95% confidence interval you will find the precise z* = 1.96. We have been using z*=2 because it is easy.

Our Example! In May 2002, the Gallup poll asked 537 randomly sampled adults the question “Generally Speaking do you believe the death penalty is applied fairly or unfairly in this country today?” Of these, 53% answered “fairly” and 7% said they didn’t know. What can we conclude from the survey?

To answer this question we’ll build a confidence interval To answer this question we’ll build a confidence interval. There are four steps: Plan, Model, Mechanics, and Conclusion.

Plan and Model Plan: Identify the parameter we want to estimate and the population you wish to make statements. Decide a confidence interval. - I want to find an interval that is likely with 95% confidence to contain the true proportion p of adults who think the penalty is applied fairly. Model: Check the conditions. Independent? Random? 10%? Success Failure? We can use a normal model.

Mechanics and Conclusion Mechanics: Construct the interval On the board Conclusion: Interpret your results. I am 95% confident that between 48.7% and 57.3% of US adults think that the death penalty is applied fairly.

Working Backwards #37 on page 449