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Calculating a confidence interval always takes more or less the same form: it involves an estimate; a multiplier; and the standard error, as below.

Normally, the multiplier is collected from a table, based on the degrees of freedom, the significance level of the confidence interval (which is oftentimes a number like 95% of 98%), and whether the confidence interval is one-sided or two-sided. Generally, we will report a two-sided range in the format and declare that we are a certain percentage certain that the actual result would be within that range. However, sometimes we are only examining possibilities that are below a certain number, or greater than a certain number, in which case we will report a one-sided range, for example, "We are 95% certain that a result would be greater than the lower bound" or "We are 95% certain that a result would be less than the upper bound."

The Bonferroni confidence interval does not drastically innovate on this approach. However, it does adjust for the effect of working with multiple variables that may be influencing one another. It is named after Carlo Emilio Bonferroni, and was developed by Olive Jean Dunn. Details are below,

Approach

Multiplier

Description

Bonferroni confidence interval

The "Bonferroni adjustment" simply involves changing the multiplier we choose. This is useful when the variables may undergo a linear combination. It essentially involves crafting a bespoke multiplier that results in a wider confidence interval which takes the multiple variables into consideration. It is generally not possible to look the multiplier up with a t-distribution table, but it is easily achievable using statistical software or spreadsheet software.

• The degree of freedom is the usual when calculating a t-multiplier.

• The probability is the usual ( for a one-sided test; for a two-sided test) but then divided by the number of variables.

For example, for a two-sided test with a 95% significance level and four variables, we will plug in for the t-multiplier with .

Bonferroni corrected confidence interval

This involves changing the multiplier we choose, as well as processing the standard error. This is useful when the variables will not undergo a linear combination.

We choose the t-multiplier based on a significance level of