Margin of Error and Confidence Interval Estimation

A random sample of 15 observations has a mean of 20 and a standard deviation of 3.5. To estimate the population mean with 95% confidence, determine the margin of error and the confidence interval.

In this problem, you are tasked with estimating a population mean based on data from a random sample. Confidence intervals are a crucial concept in inferential statistics, allowing us to express the uncertainty around our sample estimate. A confidence interval gives a range within which we expect the true population parameter to lie, with a certain level of confidence, such as 95% in this case.

To approach this problem, you need to understand the role of the t-distribution, which is particularly useful when dealing with smaller sample sizes, typically below 30 observations, where the central limit theorem's simplifying assumptions may not fully hold. The t-distribution takes into account the additional uncertainty in estimating the population standard deviation from a sample.

The margin of error in a confidence interval is a function of the sample's standard deviation, the sample size, and the desired level of confidence, which dictates the critical value from the t-distribution. This problem employs several key skills common to inferential statistics: calculating the margin of error and setting up the confidence interval using sample statistics. It is a practical exercise in determining how sample data can infer broader population characteristics, an essential part of statistical analysis.