A social media company would like to know what proportion of users are clicking on a targeted ad in their news feed. They find that the 95% confidence interval for the proportion of users who click on the targeted ad out of a sample of users exposed to this ad is (0.62, 0.72). Which of the following statements gives a valid interpretation of the 95% level of confidence?

Question

BlackTom AI · Accepted Answer

We start by restating the setup to ensure clarity: the 95% confidence interval for the population proportion of users who click the targeted ad is (0.62, 0.72) based on a sample.
Option 1: 'There is a 95% chance that the sample proportion of users who clicked on the ad is between 0.62 and 0.72.' This statement confuses the randomness of the sample with the fixed interval. The interval is constructed from the data, but the true sample proportion is a single value; the 95% level does not pertain to the probability that the observed sample proportion lies in a particular interval after the fact.
Option 2: 'We are 95% confident that the true population proportion of users who clicked on the ad is between 0.62 and 0.72.' This wording aligns with the conventional interpretation of a confidence interval: we have used a method that would produce intervals capturing the true proportion in 95% of repeated samples. The interval (0.62, 0.72) is the result for this sample, and the population proportion is believed to lie within it with that level of confidence. It’s about long-run frequency across many samples, not a probability about the single fixed parameter itself.
Option 3: 'If the sampling procedure was repeated many times, 95% of the resulting confidence intervals would not include the true population proportion of users who clicked on the ad.' This is the opposite of what a correct confidence interval implies. If the method is correct, about 95% of those intervals would include the true proportion; only about 5% would miss it, in a properly designed procedure.
Option 4: 'If the same sampling procedure was repeated many times, 95% of the resulting sample proportions would be between 0.62 and 0.72.' This misstates the concept: confidence intervals concern the range estimates for the population parameter, not where the sample proportions themselves would fall. Sample proportions from different samples could lie anywhere in [0,1], and the interval captures the parameter, not the distribution of sample estimates relative to that fixed interval.
In summary, Option 2 correctly describes the intended interpretation of a 95% confidence interval, while Options 1, 3, and 4 misinterpret the role of sample proportions, the frequency interpretation, or what the interval is supposed to convey about the population parameter.

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