The **p-value Calculator** typically requires the user to input the following information:

**Test Statistic**: The value of the test statistic (e.g.,**t-value**,**z-value**, or**F-value**) that was calculated from the data.**Degrees of Freedom**: The number of degrees of freedom associated with the test statistic.**Significance Level**: The desired level of significance, often denoted as**α (alpha)**, which is typically set at**0.05 (5%)**or**0.01 (1%)**.

## p-value Examples Chart

p-value | Interpretation |
---|---|

p < 0.01 | Highly Significant |

0.01 ≤ p < 0.05 | Significant |

0.05 ≤ p < 0.10 | Marginally Significant |

0.10 ≤ p < 0.20 | Suggestive |

p ≥ 0.20 | Not Significant |

## p-value Formula

In the case of a **z-test** or **t-test**, the **p-value** can be calculated using the following formula:

**p-value = 2 × (1 - Φ(|z|))**

Where:

**Φ(|z|)**is the cumulative distribution function of the standard normal distribution evaluated at the absolute value of the test statistic (**z-score**or**t-statistic**).

Suppose we conducted a **t-test** and obtained a **t-statistic** of **2.45** with **20 degrees of freedom**. To calculate the **p-value**, we would use the formula:

**p-value = 2 × (1 - Φ(|2.45|))**

Where **Φ(|2.45|)** is the cumulative distribution function of the standard normal distribution evaluated at the absolute value of the **t-statistic**, which is **2.45**.

Using a standard normal distribution table or calculator, we can find that **Φ(2.45) ≈ 0.9928**.

**p-value = 2 × (1 - 0.9928) = 0.0144**

Therefore, the

p-valuefor thist-testwith at-statisticof2.45and20 degrees of freedomis approximately0.0144.

## How is the p-value Calculated?

The **p-value** is calculated based on the **test statistic** (e.g., **z-statistic**, **t-statistic**, or **F-statistic**) and the underlying probability distribution of the test statistic under the **null hypothesis**.

Suppose we want to test the hypothesis that the mean of a population is equal to a certain value (the **null hypothesis**, **H0**). We take a sample from the population, calculate the sample mean, and then compute the **t-statistic**:

**t = (Sample Mean - Hypothesized Mean) / (Standard Error of the Sample Mean)**

The **t-statistic** follows a **t-distribution** with **(n-1)** degrees of freedom, where **n** is the sample size.

The **p-value** is the probability of obtaining a **t-statistic** at least as extreme as the one observed, given that the **null hypothesis** is true. In other words, it represents the likelihood of seeing the observed (or more extreme) results if the **null hypothesis** is correct.

To calculate the **p-value**, we follow these steps:

- Determine the appropriate probability distribution (in this case, the
**t-distribution**) based on the test statistic and the degrees of freedom. - Calculate the absolute value of the
**t-statistic**,**|t|**. - Use the cumulative distribution function (CDF) of the
**t-distribution**to find the probability of obtaining a**t-statistic**with an absolute value greater than or equal to**|t|**. This is the**p-value**.

The calculated **t-statistic** is **2.45** and the degrees of freedom are **20**. Using a **t-distribution** table or calculator, we can find the **p-value**:

**p-value = 2 × (1 - Φ(|2.45|))**

Where **Φ(|2.45|)** is the CDF of the standard normal distribution evaluated at the absolute value of the **t-statistic**, which is **2.45**.

**p-value = 2 × (1 - 0.9928) = 0.0144**

**The p-value for this t-test with a t-statistic of 2.45 and 20 degrees of freedom is approximately 0.0144 or 1.44%.**