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Data-Centric Statistical Inference Using R and Tidyverse

Precise and Shorthand Interpretation

Understand the precise and common shorthand interpretations of 95% confidence intervals and how they reflect the reliability of statistical inference. Explore how confidence levels impact interval width and how larger sample sizes lead to more precise, narrower intervals. This lesson equips you to interpret confidence intervals properly and appreciate factors influencing their construction.

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Let’s return our attention to 95% confidence intervals. The precise and mathematically correct interpretation of a 95% confidence interval is a little long-winded:

  • Precise interpretation: If we repeated our sampling procedure a large number of times, we expect about 95% of the resulting confidence intervals to capture the value of the population parameter.

This is what we observed. Our confidence interval construction procedure is 95% reliable. That is to say, we can expect our confidence intervals to include the true population parameter about 95% of the time.

A common but incorrect interpretation is that there’s a 95% probability that the confidence interval contains pp. Each of the confidence intervals either does or doesn’t contain pp. In other words, the probability is either a 1 or a 0.

So, if the 95% confidence level only relates to the reliability of the confidence interval construction procedure and not to a given confidence interval itself, what insight can be derived from a given confidence interval? For example, going back to the pennies example, we found that the percentile method 95% confidence interval for μ\mu was (1991, 1999), whereas the standard error method 95% confidence interval was (1991, 1999). What can be said about these two intervals?

Loosely ...