### Introduction

- In a hypothesis test problem, you may see words such as,
*the level of significance is 1 percent*. The*1 percent*is the preconceived or preset*α*. - The statistician setting up the hypothesis test selects the value of
*α*to use*before*collecting the sample data. *If no level of significance is given, a common standard to use is**α*= 0.05.- When you calculate the
*p*-value and draw the picture, the*p*-value is the area in the left tail, the right tail, or split evenly between the two tails. For this reason, we call the hypothesis test left, right, or two tailed. - The
**alternative hypothesis**, ${H}_{a}$, tells you if the test is left, right, or two-tailed. It is the*key*to conducting the appropriate test. *H*has a symbol that contains an equal sign._{a}never*Thinking about the meaning of the**p*-value: A data analyst should have more confidence that he made the correct decision to reject the null hypothesis with a smaller*p*-value (for example, 0.001 as opposed to 0.04) even if using the 0.05 level for alpha. Similarly, for a large*p*-value such as 0.4, as opposed to a*p*-value of 0.056 (alpha = 0.05 is less than either number), a data analyst should have more confidence that she made the correct decision in not rejecting the null hypothesis. This makes the data analyst use judgment rather than mindlessly applying rules.

The following examples illustrate a left-, right-, and two-tailed test.

### Example 9.11

*H _{0}*:

*μ*= 5

*H*:

_{a}*μ*< 5

Test of a single population mean. *H _{a}* tells you the test is left-tailed. The picture of the

*p*-value is as follows:

*H _{0}*:

*μ*= 10

*H*:

_{a}*μ*< 10

Assume the *p*-value is 0.0935. What type of test is this? Draw the picture of the *p*-value.

### Example 9.12

*H _{0}*:

*p*≤ 0.2

*H*:

_{a}*p*> 0.2

This is a test of a single population proportion. *H _{a}* tells you the test is

**right-tailed**. The picture of the

*p*-value is as follows:

*H _{0}*:

*μ*≤ 1

*H*:

_{a}*μ*> 1

Assume the *p*-value is 0.1243. What type of test is this? Draw the picture of the *p*-value.

### Example 9.13

*H _{0}*:

*p*= 50

*H*:

_{a}*p*≠ 50

This is a test of a single population mean. *H _{a}* tells you the test is

**two-tailed**. The picture of the

*p*-value is as follows.

*H _{0}*:

*p*= 0.5

*H*:

_{a}*p*≠ 0.5

Assume the *p*-value is 0.2564. What type of test is this? Draw the picture of the *p*-value.