## Confidence intervals

 Confidence Interval Confidence Limits Confidence Level Confidence Interval for a Mean Confidence Interval for the Difference Between Two Means

Confidence Interval

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.

If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter. Confidence intervals are usually calculated so that this percentage is 95%, but we can produce 90%, 99%, 99.9% (or whatever) confidence intervals for the unknown parameter.

The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter (see precision). A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter.

Confidence intervals are more informative than the simple results of hypothesis tests (where we decide "reject H0" or "don't reject H0") since they provide a range of plausible values for the unknown parameter.

Confidence Limits

Confidence limits are the lower and upper boundaries / values of a confidence interval, that is, the values which define the range of a confidence interval.

The upper and lower bounds of a 95% confidence interval are the 95% confidence limits. These limits may be taken for other confidence levels, for example, 90%, 99%, 99.9%.

Confidence Level

The confidence level is the probability value associated with a confidence interval.

It is often expressed as a percentage. For example, say , then the confidence level is equal to (1-0.05) = 0.95, i.e. a 95% confidence level.

Example
Suppose an opinion poll predicted that, if the election were held today, the Conservative party would win 60% of the vote. The pollster might attach a 95% confidence level to the interval 60% plus or minus 3%. That is, he thinks it very likely that the Conservative party would get between 57% and 63% of the total vote.

Confidence Interval for a Mean

A confidence interval for a mean specifies a range of values within which the unknown population parameter, in this case the mean, may lie. These intervals may be calculated by, for example, a producer who wishes to estimate his mean daily output; a medical researcher who wishes to estimate the mean response by patients to a new drug; etc.

The (two sided) confidence interval for a mean contains all the values of 0 (the true population mean) which would not be rejected in the two-sided hypothesis test of:
H0: µ = µ0
against
H1: µ not equal to µ0

The width of the confidence interval gives us some idea about how uncertain we are about the unknown population parameter, in this cas the mean. A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter.

We calculate these intervals for different confidence levels, depending on how precise we want to be. We interpret an interval calculated at a 95% level as, we are 95% confident that the interval contains the true population mean. We could also say that 95% of all confidence intervals formed in this manner (from different samples of the population) will include the true population mean.

Compare one sample t-test.

Confidence Interval for the Difference Between Two Means

A confidence interval for the difference between two means specifies a range of values within which the difference between the means of the two populations may lie. These intervals may be calculated by, for example, a producer who wishes to estimate the difference in mean daily output from two machines; a medical researcher who wishes to estimate the difference in mean response by patients who are receiving two different drugs; etc.

The confidence interval for the difference between two means contains all the values of µ1 - µ2 (the difference between the two population means) which would not be rejected in the two-sided hypothesis test of:
H0: µ1 = µ2
against
H1: µ1 not equal to µ2
i.e.
H0: µ1 - µ2 = 0
against
H1: µ1 - µ2 not equal to 0

If the confidence interval includes 0 we can say that there is no significant difference between the means of the two populations, at a given level of confidence.

The width of the confidence interval gives us some idea about how uncertain we are about the difference in the means. A very wide interval may indicate that more data should be collected before anything definite can be said.

We calculate these intervals for different confidence levels, depending on how precise we want to be. We interpret an interval calculated at a 95% level as, we are 95% confident that the interval contains the true difference between the two population means. We could also say that 95% of all confidence intervals formed in this manner (from different samples of the population) will include the true difference.

Compare two sample t-test.