4.3 Summary

A confidence interval gives a range of most plausible values for a parameter $\theta$.

A confidence interval should be interpreted using repeated sampling ideas: if 100 independent samples each of size $n$ were taken from the population, and a 95% confidence interval for a parameter was estimated using each of these samples, then we would expect 95 of the confidence intervals to contain the true parameter value.

A related concept is the idea of a bootstrap confidence interval:

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  We create a set of pseudo-samples from our actual sample by sampling with replacement.

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  For each pseudo (or bootstrapped) sample we estimate the parameter. Looking across all parameter estimates gives an idea of the sampling distribution for the parameter.

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  The 95% confidence interval is obtained by taking the 2.5% and 97.5% quantiles of the parameter estimates.

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  The 90% confidence interval is obtained by taking the 5% and 95% quantiles of the parameter estimates.

In some cases it is possible to use a statistical model for the data to derive a model-based confidence interval. In the case of the population mean $\mu$ of an i.i.d sample of continuous data, the $100(1-\alpha )\%$ confidence interval is given by

We can use confidence intervals to test hypotheses, by assessing whether the value of the parameter under $H_0$ lies inside (accept $H_0$) or outside (reject $H_0$) an appropriate confidence interval.

We have calculated confidence intervals for the difference between two means, for both paired and unpaired data. These confidence intervals can also be used to test whether or not the means of the two populations are equal.