Final answer:
To construct accurate confidence intervals for a population mean with an unknown standard deviation, statisticians use the Student's t-distribution, which accounts for the extra variability from estimating the population standard deviation with the sample standard deviation, especially in small sample sizes.
Step-by-step explanation:
When developing a confidence interval for a population mean, statisticians must adjust their methods based on whether the population standard deviation is known. If the population standard deviation, denoted as σ, is known, we can use the normal distribution to compute the confidence interval. However, when σ is unknown, which is often the case in practice, we estimate it using the sample standard deviation, s. The sample mean, x, also serves as a point estimate for the population mean, μ.
With small sample sizes, using the sample standard deviation s in place of σ can cause inaccuracies in the confidence interval. To address this, the Student's t-distribution is used rather than the normal distribution. The Student’s t-distribution accounts for the additional variability introduced by estimating σ with s, and it provides a more reliable method for constructing confidence intervals when the sample size is small.
The studentized version of the sample mean, often represented as a t-score, is calculated using the formula t = ( x - μ ) / ( s / √n ), where x is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
By taking these factors into consideration and using the t-score for calculating confidence intervals, we can interpret the interval as a range where there is a specified level of confidence that it contains the true population mean. This is crucial for accurate statistical inference, especially when working with small sample sizes and an unknown population standard deviation.