Confidence Interval for the Mean (small sample)

A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. It is used to estimate the uncertainty around a sample statistic.

A 99% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 99 of the intervals would contain the true population mean.

The formula is: CI = x̄ ± t*(s/√n), where x̄ is the sample mean, t is the t-score from the t-distribution, s is the sample standard deviation, and n is the sample size.

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used when the sample size is small and the population standard deviation is unknown.

The t-score can be found using a t-table or calculator, based on the desired confidence level and the degrees of freedom (df = n - 1, where n is the sample size).

Larger sample sizes generally lead to narrower confidence intervals, indicating more precise estimates of the population parameter.

As the confidence level increases, the width of the confidence interval also increases. This is because a higher confidence level requires a larger range to ensure that the true parameter is captured.

Standard deviation measures the amount of variation or dispersion in a set of values. In confidence intervals, it helps determine the spread of the data around the mean.

A confidence interval provides a range of values that is believed to contain the population mean. For example, a 95% CI of (2.5, 3.2) suggests we are 95% confident that the true mean lies within this range.

Outliers can significantly affect the mean and standard deviation, leading to wider confidence intervals and potentially misleading conclusions about the population parameter.