Introduction to Statistics: Confidence Intervals

Question

I am in an Intro to Stats class, and I need help with this problem:

According to the Pew Research Foundation, based on a random sample of $1,001$ adults, a $95\%$ confidence interval for the proportion of adults who would ride in a driverless car is $(0.45,0.51).$

What is the best interpretation of this confidence interval?

a) We are $95\%$ confident that the population proportion is between $0.45$ and $0.51?$

b)We are $95\%$ confident that the sample proportion is between $0.45$ and $0.51?$

I am currently leaning towards A since C.I. are population proportion and the professor did not really go in depth over the distinction between the two. I am wondering if there is a case where we would have sample, or is it always proportion?

Answer 1

I think this is the most concise and nice definition of what CI's mean:

A confidence interval for a population parameter, $\theta$, is a random interval, calculated from the sample, that contains $\theta$ with some specified probability. For example, a $95\%$ CI for $\mu$ is a random interval that contains $\mu$ with probability $0.95$; if we were to take many random samples and form a confidence interval from each one, about $95\%$ of these intervals would contain $\mu$.

                                            Mathematical Statistics and Data Analysis
                                                                         John A. Rice

Of note, there is a subtlety not captured in either one of the answers in your test: Notice that what it really means is that if we were to repeat the sampling process an infinity number of times, the true population parameter would be included in the CI in $95\%$ of the occasions.

This is different than an assessment of of confidence.

However, and given the binary choice you face, it is clear that you are aiming at estimating the population parameter.

Answer 2

Yes, (a) is the answer they are looking for. One good way to know it can't be (b) is that they already know the sample proportion exactly (it's just the proportion in their data). Confidence intervals are an inference about population proportions based on a sample.

More exactly, 95% of the time a confidence interval is derived from a random sample of size N, it will cover the population proportion.