# 7. The Relationship between Confidence Intervals and Tests of Significance

## d. explain a decision rule, the power of a test, and the relation between confidence intervals and hypothesis tests;

What is a confidence interval?

A confidence interval is a range of values within which it is believed that a certain unknown population parameter (often the mean) will fall with a certain degree of confidence. The percentage of confidence is denoted `(1 - \alpha)%`

.

What is a test of significance?

A test of significance is a test to ascertain whether or not the value of an unknown population parameter is as stated by an individual or institution. This test is carried out at a significance level of `\alpha`

, which determines the size of the rejection region.

What is a test of significance testing for? What is the math formula?

- We are testing to see the likelihood that the z-value will be less than the test statistic.

- $$\mu_0 < \bar{x} - \frac{(z_{a/2})(\sigma)}{\sqrt{n}}$$

When conducting a significance test and the confidence interval contains the value of the unknown population parameter as hypothesized in the null hypothesis (`H_0`

), is the null hypothesis rejected or not?

Not rejected

Given the following info, is the p-value > or < 0.05?

`H_0:\mu = 50`

, `H_a:\mu != 50`

, normal population with `\sigma = 6`

. Standard confidence interval for the mean is (51.3, 54.7).

##### Source:

- CFA