95% confidence interval

Statistical range of values within which a parameter is expected to lie

95% Confidence Interval (CI) is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval. The 95% confidence level means that if the same population is sampled multiple times, approximately 95% of the intervals calculated from those samples will contain the true population parameter.

Calculation

The 95% confidence interval is calculated using the formula:

CI = \( \bar{x} \pm Z \frac{\sigma}{\sqrt{n}} \)

where:

• \( \bar{x} \) is the sample mean,
• \( Z \) is the Z-score from the standard normal distribution for a 95% confidence level (approximately 1.96),
• \( \sigma \) is the population standard deviation, and
• \( n \) is the sample size.

In cases where the population standard deviation is unknown, the sample standard deviation \( s \) is used, and the t-distribution is applied instead of the normal distribution.

Interpretation

A 95% confidence interval provides a range of plausible values for the population parameter. It is important to note that the interval does not imply that there is a 95% probability that the parameter lies within the interval. Instead, it means that if we were to take many samples and build a confidence interval from each sample, 95% of those intervals would contain the true parameter.

Assumptions

The calculation of a 95% confidence interval assumes that:

• The data is randomly sampled from the population.
• The data follows a normal distribution, or the sample size is large enough for the Central Limit Theorem to apply.
• The samples are independent of each other.

Applications

Confidence intervals are widely used in various fields such as medicine, psychology, and social sciences to estimate population parameters like means, proportions, and differences between groups. In clinical trials, for example, confidence intervals are used to estimate the effect size of a treatment.

Limitations

While confidence intervals provide valuable information about the precision of an estimate, they have limitations:

• They do not account for systematic errors or biases in the data.
• They rely on the assumption that the data is normally distributed.
• The interpretation of confidence intervals can be misunderstood, leading to incorrect conclusions.

Also see

• Confidence interval
• Standard deviation
• Central Limit Theorem
• Normal distribution
• T-distribution