$H_o$: | μ₀ | ||

$H_a$: | μ | < > ≠ | μ₀ |

$n$ | = | $\bar{x}$ | = | = |

$\text{Test Statistic: }$ | = |

$\text{Degrees of Freedom: } $ | $df$ | = |

$ \text{Level of Significance: } $ | $\alpha$ | = |

$H_o$: | $\mu$ | ||

$H_a$: | $\mu$ | < > ≠ | $\mu_0$ |

$n$ | = | σ | = | $\mu$ | = |

$\text{Level of Significance: }$ | $\alpha$ | = |

Example 1 • Example 2

Example 1 • Example 2

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

$\sigma$ Known | $\sigma$ Unknown | |

Test Statistic | $ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $ | $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $ |

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H_{a}). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

Lower Tail Test | Upper Tail Test | Two-Tailed Test |

$H_0 \colon \mu \geq \mu_0$ | $H_0 \colon \mu \leq \mu_0$ | $H_0 \colon \mu = \mu_0$ |

$H_a \colon \mu < \mu_0$ | $H_a \colon \mu > \mu_0$ | $H_a \colon \mu \neq \mu_0$ |

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

Lower Tail Test | Upper Tail Test | Two-Tailed Test |

If $z \leq -z_\alpha$, reject $H_0$. | If $z \geq z_\alpha$, reject $H_0$. | If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$. |

If $t \leq -t_\alpha$, reject $H_0$. | If $t \geq t_\alpha$, reject $H_0$. | If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$. |

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Condition | ||||

$H_0$ True | $H_a$ True | |||

Conclusion | Accept $H_0$ | Correct | Type II Error | |

Reject $H_0$ | Type I Error | Correct |

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator. The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator. The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.