Question

Explain how the ideas of hypothesis testing and rejecting the null hypothesis can be related to confidence interval. Be sure to also consider the idea of one-sided confidence intervals.

Answer 1

There is an extremely close relationship between confidence intervals and hypothesis testing. When a 95% confidence interval is constructed, all values in the interval are considered plausible values for the parameter being estimated. Values outside the interval are rejected as relatively implausible. If the value of the parameter specified by the null hypothesis is contained in the 95% interval then the null hypothesis cannot be rejected at the 0.05 level. If the value specified by the null hypothesis is not in the interval then the null hypothesis can be rejected at the 0.05 level. If a 99% confidence interval is constructed, then values outside the interval are rejected at the 0.01 level.

Confidence intervals and hypothesis tests are similar in that they are both inferential methods that rely on an approximated sampling distribution. Confidence intervals use data from a sample to estimate a population parameter. Hypothesis tests use data from a sample to test a specified hypothesis. Hypothesis testing requires that we have a hypothesized parameter.

Confidence intervals contain a range of reasonable estimates of the population parameter. If the confidence intervals constructed were two-tailed. These two-tailed confidence intervals go hand-in-hand with the two-tailed hypothesis tests. The conclusion drawn from a two-tailed confidence interval is usually the same as the conclusion drawn from a two-tailed hypothesis test. In other words, if the 95% confidence interval contains the hypothesized parameter, then a hypothesis test at the 0.05 α level will almost always fail to reject the null hypothesis. If the 95% confidence interval does not contain the hypothesize parameter, then a hypothesis test at the 0.05 α level will almost always reject the null hypothesis.

One-tailed hypothesis tests are also known as directional and one-sided tests because you can test for effects in only one direction. When you perform a one-tailed test, the entire significance level percentage goes into the extreme end of one tail of the distribution.

We can construct one-sided confidence intervals with 95% coverage.

The two-sided confidence interval corresponds to the critical values in a two-tailed hypothesis test, the same applies to one-sided confidence intervals and one-tailed hypothesis tests.

If the one-sided confidence interval has values outside the null hypothesis in the one-sided test then we reject Ho.

For e.g.

1) If we want to test,

Ho:- mu = 0 vs

H1:- mu != 0

And two-sided confidence interval is (0.2,3)

Here two-sided confidence interval is outside null value (0 in the null hypothesis) so we reject Ho.

If two sided confidence interval is (-2 , 1) then it have o in it we have to fail to reject Ho.

2) If we want to test,

Ho:- mu = 0    vs

H1:- mu > 0

And one sided confidence interval is (0.2 ,3)

Here one confidence interval is outside null value (0 in null hypothesis) so we reject Ho.

If one sided confidence interval is (-2, 1) then it have 0 in it we have to fail to reject Ho.

3) If we want to test,

Ho:- mu = 0    vs

H1:- mu < 0

And one sided confidence interval is (-2,-1)

Here one confidence interval is outside null value (0 in null hypothesis) so we reject Ho.

If one sided confidence interval is (-2, 3) then it have 0 in it we have to fail to reject Ho.