Think Stats Exploratory Data Analysis in Python

Another way to summarize the sampling distribution is to compute a confidence interval. For example, a 90% confidence interval contains 90% of the values in the sampling distribution, which we can find by computing the 5th and 95th percentiles. Here’s the 90% confidence interval for the average weight of chinstrap penguins.

$ ci90 = np.percentile(sample_means, [5, 95])
ci90
array([3.6576334 , 3.80737506])

To interpret a confidence interval, it is tempting to say that there is a 90% chance that the true value of the population parameter falls in the 90% confidence interval. In this example, we would say there is a 90% chance that the population mean for chinstrap penguins is between 3.66 and 3.81 kg.

Under a strict philosophy of probability called frequentism, this interpretation would not be allowed, and in many statistics books, you will be told that this interpretation is wrong.

In my opinion, this prohibition is unnecessarily strict. Under reasonable philosophies of probability, a confidence interval means what people expect it to mean: there is a 90% chance that the true value falls in the 90% confidence interval.

However, confidence intervals only quantify variability due to sampling -- that is, measuring only part of the population. The sampling distribution does not account for other sources of error, notably sampling bias and measurement error, which are the topics of the next section.