What is the meaning of 95% confidence interval?

As a statistical expert with a strong background in data analysis and interpretation, I often encounter the concept of confidence intervals in my work. They are a fundamental tool in statistical inference, used to estimate the range within which a population parameter is likely to fall. Let's delve into the meaning of a 95% confidence interval.
A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. The 95% confidence interval specifically refers to the level of confidence that the true population parameter lies within this range. It is a statistical expression of the uncertainty associated with an estimate of a population parameter based on a sample of data.
The concept of a confidence interval is closely related to the idea of a sampling distribution. When we take a sample from a population, we can calculate an estimate for a parameter such as the mean. If we were to take many different samples from the same population and calculate the mean for each, we would end up with a distribution of these means, which is the sampling distribution of the mean. The confidence interval is based on the assumption that the sample mean will be approximately normally distributed if the sample size is large enough, even if the population distribution is not normal (this is known as the Central Limit Theorem).
Now, let's address the common misconceptions and clarify the meaning of a 95% confidence interval with the help of the provided information:
1. **"If repeated samples were taken and the 95% confidence interval was computed for each sample, 95% of the intervals would contain the population mean."** This statement is a correct interpretation of what a 95% confidence interval represents. It means that if we were to take an infinite number of samples, calculate a confidence interval for each, and then look at all those intervals, we would expect that 95% of them would include the true population mean. It's important to note that this is a property of the method of interval estimation, not a guarantee for any single interval.
2. **"A 95% confidence interval has a 0.95 probability of containing the population mean."** This is a correct statement. It reflects the probabilistic nature of confidence intervals. It's not saying that there's a 95% chance the population mean is within a particular interval we've calculated; rather, it's saying that if we were to construct many such intervals using the same method, 95% of them would contain the true population mean.
3. **"95% of the population distribution is contained in the confidence interval."** This statement is incorrect. A confidence interval does not imply that 95% of the population lies within the interval. Instead, it's about the parameter estimate itself. It's the interval that has a 95% chance of capturing the true parameter, not that it spans 95% of the population's values.
In summary, a 95% confidence interval is a range that, when calculated from a sample, provides an estimate that is likely to include the true population parameter 95% of the time. It's a measure of the reliability of an estimate, reflecting the precision of the sample-based estimate rather than the spread of the data within the population. It's crucial to understand that a confidence interval provides a range of plausible values for the parameter, not a statement about the proportion of the population that falls within that range.

If repeated samples were taken and the 95% confidence interval was computed for each sample, 95% of the intervals would contain the population mean. A 95% confidence interval has a 0.95 probability of containing the population mean. 95% of the population distribution is contained in the confidence interval.