Confidence Interval for Estimating Population Mean

A point estimate from a single sample often fails to represent the population well because it does not account for population variability. Even when samples are randomly selected, point estimates can be biased. This unreliability necessitates using a confidence interval, which provides a range of values to better estimate the true population mean.

When population standard deviation is known, the z distribution determines the critical value. When it is unknown, the t distribution is used instead. Confidence intervals calculated using the t distribution are wider than those using z values, but both are equally accurate in predicting the population mean. Not knowing the population standard deviation does not make estimates inaccurate.

A 95% confidence level means that if many confidence intervals are obtained using identical sampling methods, 95% of them will contain the true population mean. It does not mean there is a 95% chance the population mean falls within any single calculated interval, since the population mean is a fixed value. This distinction is critical for proper interpretation of confidence intervals.

When population standard deviation is unknown and the t distribution is used, samples must be drawn from a normally distributed population, or the sample size should exceed 30. The confidence interval depends on degrees of freedom, which relate directly to sample size. These requirements ensure reliable margin of error calculations.

The t distribution is non-normal and symmetric, reflecting greater uncertainty when population standard deviation is unknown. This additional uncertainty is reflected in wider confidence intervals compared to those calculated with known standard deviation using the z distribution. Despite the wider range, both methods provide equally accurate estimates of the population mean.

Calculating a confidence interval requires a margin of error and prior knowledge of either the population standard deviation or variance. The margin of error is computed differently depending on whether standard deviation is known or unknown. These components combine with a predetermined confidence level, such as 90%, 95%, or 99%, to establish the interval range.

The t distribution confidence interval depends on degrees of freedom, which are determined by sample size. As sample size increases, the t distribution approaches the z distribution. This relationship means that larger samples produce narrower confidence intervals and more precise estimates of the population mean.