95% CI

95% Confidence Interval (CI) is a term widely used in statistics to describe the range of values within which we can be 95% certain that the true value (for example, the true mean of a population) lies. It is a key concept in statistical inference, allowing researchers to make predictions and decisions based on sample data without having access to the entire population.

Definition

A 95% CI is calculated from the sample data and is used to estimate the range of values that are likely to include the population parameter. The "95%" part of the term indicates that if we were to take many samples and build a confidence interval from each sample, approximately 95% of these intervals would contain the true population parameter.

Calculation

The calculation of a 95% CI depends on the standard deviation of the sample, the sample size, and the sample mean. The formula for a 95% CI of a mean is typically:

\[CI = \bar{x} \pm (t_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}})\]

where: - \(\bar{x}\) is the sample mean, - \(t_{\frac{\alpha}{2}}\) is the t-value from the t-distribution table corresponding to the desired confidence level and degrees of freedom (n-1), - \(s\) is the sample standard deviation, and - \(n\) is the sample size.

Interpretation

Interpreting a 95% CI requires understanding that the interval has a 95% probability of containing the true population parameter. However, this does not mean that there is a 95% chance that any given interval contains the parameter; the interval either does or does not contain the parameter. The "95%" refers to the long-run success rate of the method used to construct the interval.

Applications

95% CIs are widely used in various fields such as medicine, biology, psychology, and economics to estimate parameters like means, proportions, and differences between means. They are crucial in hypothesis testing, where they can provide more information than simple p-values, such as the size and direction of an effect.

Limitations

While 95% CIs are useful, they have limitations. They are sensitive to outliers and non-normal data distributions. Additionally, the width of the interval depends on the sample size, with smaller samples leading to wider intervals. This means that the precision of the estimate can vary significantly depending on the sample size.

See Also

• Statistical significance
• P-value
• Standard deviation
• Sample size determination

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