- Explain how a confidence interval is related to a two-sided hypothesis test.
Confidence Interval – Hypothesis Testing
A confidence interval provides a range of population values with which a sample statistic is consistent at a given confidence level. In many cases, a confidence interval can also be used to either reject or not reject the null hypothesis, and therefore perform the same function as the typical hypothesis test.
The Lego experiment considered the population parameter of the difference of proportions, [latex]p_{1} -p_{2}[/latex]. The confidence intervals constructed were two-tailed, since [latex]-z^{*}[/latex] is the point on the standard normal distribution such that the proportion of area under the curve between [latex]-z^{*}[/latex] and [latex]+z^{*}[/latex] is [latex]C[/latex], the confidence level.
For example, a [latex]95\%[/latex] confidence interval corresponds to a hypothesis test with a significance level of [latex]5\%[/latex], or [latex]\alpha = 0.05[/latex]. Similarly, a [latex]99\%[/latex] confidence interval corresponds to a hypothesis test with a significance level of [latex]1\%[/latex], or [latex]\alpha =0.01[/latex].
Let’s connect the result of the confidence interval to the conclusion of the hypothesis test.
Typically, the conclusion drawn from a two-tailed confidence interval is usually the same as the conclusion drawn from a two-tailed hypothesis test.
- If a confidence interval contains the hypothesized parameter, a hypothesis test at the [latex]0.05[/latex] level will almost always fail to reject the null hypothesis.
- If the [latex]95\%[/latex] confidence interval does not contain the hypothesized parameter, a hypothesis test at the [latex]0.05[/latex] level will almost always reject the null hypothesis.
While this does not always hold for tests of proportions, a confidence interval typically provides more information about reasonable values of the parameter.