Chapter 7: Estimating Parameters and Determining Sample Sizes

The chapter begins with point estimates, which are single numerical values derived from sample data that serve as best guesses for unknown population parameters like the mean or proportion. However, point estimates alone provide no information about their reliability, so the chapter introduces confidence intervals as a more informative approach that establishes a range of plausible values for a parameter along with a specified level of certainty. Students learn to construct confidence intervals for population means under two scenarios: when the population standard deviation is known (using the normal distribution and z-critical values) and when it is unknown (using the t-distribution and t-critical values). The chapter explains why the t-distribution becomes necessary when working with smaller samples or unknown population variability, and how to locate appropriate critical values using statistical tables or technology. Beyond means, the chapter covers methods for estimating population proportions and population standard deviations or variances, each with distinct formulas and distributional assumptions. A critical component is the concept of margin of error, which quantifies the maximum expected difference between a point estimate and the true parameter value at a given confidence level. The chapter then shifts to sample size determination, providing formulas that allow researchers to calculate how many observations are needed to achieve a desired margin of error for both mean and proportion estimation. This section highlights the practical trade-off between statistical precision and resource constraints, recognizing that larger samples increase accuracy but require more time and expense. Throughout, the chapter emphasizes proper interpretation of confidence levels and intervals, addressing common misunderstandings such as treating a confidence interval as a probability statement about the parameter itself rather than a statement about the long-run behavior of the estimation procedure. Real-world applications in health research, market analysis, and opinion polling demonstrate how these techniques ensure that statistical conclusions are both meaningful and defensible.