A 95% confidence interval is a statistical concept used to infer the reliability of an estimate. Here’s how it relates to Washington’s population data:

Confidence Interval (CI): This is a range of values, derived from sample statistics, that is believed to contain the true population parameter (like the mean or variance) with a certain level of confidence. For Washington’s population data, the CI gives us a range that we are 95% confident contains the true population mean or variance.

Level of Certainty: The 95% level indicates that if we were to take many samples and construct a confidence interval from each of them, we would expect 95% of those intervals to contain the true population parameter. It’s not a guarantee for a single interval, but it gives a high level of certainty over the long run.

Estimates and True Values: While the interval gives an estimate range, it does not specify the exact values of the population parameters. The true mean and variance of Washington’s population might lie anywhere within this interval, and there’s a 5% chance they could lie outside of it.

Ambiguity or Imprecision: Despite its utility, the confidence interval does not eliminate uncertainty. There’s always a 5% likelihood that even our best statistical methods might miss the true parameter. This is not due to a flaw in the calculation but rather is a consequence of the variability inherent in any sample data.

Statistical Inference: The CI is an example of statistical inference, which involves making judgments about a population based on sample data. In this case, we are inferring the possible range of the population mean or variance for Washington’s population based on a sample drawn from that population.

Practical Use: Confidence intervals are often used in policy-making, scientific research, and various forms of analysis. For instance, if we are looking at average household income in Washington, a 95% confidence interval would help policymakers understand the variation and uncertainty in income estimates, aiding in better decision-making.

To construct a 95% confidence interval, statisticians use sample data to calculate the interval’s lower and upper bounds, typically applying the formula that includes the sample mean, the standard deviation, and the Z-score or T-score corresponding to the desired level of confidence (which is 1.96 for 95% confidence with a large sample size and normal distribution, or a T-score for smaller samples or unknown population standard deviation).

Understanding that these intervals are built on probability and are subject to sample variability is key when interpreting their meaning in real-world applications.