Confidence Intervals

Understanding Confidence Intervals

A confidence interval is a range of values, derived from the statistical analysis of the given data, which is likely to contain the true value of an unknown population parameter.
The confidence level represents the frequency (in percentage) at which the range of values encapsulates the true parameter in repeated sampling.
The term “confidence” does not imply that the interval necessarily contains the true value; instead, it expresses our confidence in the method used to construct the interval.
Each confidence interval is computed from a particular sample, and different samples give different intervals.

The construction of a confidence interval depends on several factors, including the sample mean or proportion, the variability in the data (as indicated by the standard deviation), the sample size, and the desired confidence level.
For large sample sizes (n > 30) or if the population is known to be normally distributed, we can use the z-score in the calculation of the confidence interval.
When the sample size is smaller (n ≤ 30) and the population standard deviation is unknown, we use the t-distribution instead of the z-distribution.
The formula to calculate a z-confidence interval for a population mean (μ) is: x̄ ± z(σ/√n), where x̄ is the sample mean, z is the z-score for the desired confidence level, σ is the population standard deviation, and n is the sample size.
A confidence interval for a population proportion (p) can be calculated using the formula: p̂ ± z(√[(p̂(1 - p̂))/n]), where p̂ is the sample proportion and n is the sample size.

If a 95% confidence interval for a parameter includes a certain value (say, zero for effect size), we may conclude at the 5% significance level that there is no statistically significant difference or effect.
It’s important to interpret confidence intervals correctly, not as a range of plausible values for the parameter, but as an indicator of the method’s precision.
A smaller confidence interval indicates a higher precision of the estimate, while a wider interval signifies less precision. However, a smaller interval does not ensure that the interval contains the true parameter value.
If confidence intervals for two groups do not overlap, this is a visual sign that the difference between the groups is likely to be statistically significant.

When conducting hypothesis testing at a significance level α, if the null value falls within the 100(1 - α)% confidence interval, we would not reject the null hypothesis.
Conversely, if the null value falls outside this interval, we reject the null hypothesis at the α level of significance.
It is a common mistake to confuse the confidence level with the significance level (α) - they are not the same. The confidence level is equivalent to 1 - α, i.e., it represents the likelihood of not rejecting the null hypothesis when it is true.
Understand that hypothesis testing is about testing a claim, while confidence intervals provide a range of plausible values for an unknown parameter.