### 9.1 Null and Alternative Hypotheses

*H _{0}* and

*H*are contradictory.

_{a}If H has:_{0} |
equal (=) | greater than or equal to (≥) | less than or equal to (≤) |

then H has:_{a} |
not equal (≠) or greater than (>) or less than (<) |
less than (<) | greater than (>) |

If *α* ≤ *p*-value, then do not reject *H _{0}*.

If *α* > *p*-value, then reject *H _{0}*.

*α* is preconceived. Its value is set before the hypothesis test starts. The *p*-value is calculated from the data.

### 9.2 Outcomes and the Type I and Type II Errors

*α* = probability of a Type I error = *P*(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

*β* = probability of a Type II error = *P*(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

### 9.3 Distribution Needed for Hypothesis Testing

If there is no given preconceived *α*, then use *α* = 0.05.

Types of Hypothesis Tests

- Single population mean,
*known*population variance (or standard deviation):**Normal test**. - Single population mean,
*unknown*population variance (or standard deviation):**Student's**.*t*-test - Single population proportion:
**Normal test**. - For a
*single population mean*, we may use a normal distribution with the following mean and standard deviation. Means: $\mu ={\mu}_{\overline{x}}$ and ${\sigma}_{\overline{x}}=\frac{{\sigma}_{x}}{\sqrt{n}}\text{.}$ - For a
*single population proportion*, we may use a normal distribution with the following mean and standard deviation. Proportions:*µ*=and $\sigma =\sqrt{\frac{pq}{n}}$.*p*