p-value

1. Overview

Probability of obtaining a sample "more extreme" than the ones observed in your data, assuming $H_{0}$ is true

The p-value is one of the key outputs of analyzing data. The p-value is the probability that, if the null hypothesis were true, sampling variation would produce an estimate that is further away from the hypothesized value than our data estimate. Shortly, The p-value tells us how likely it is to get a result like this if the Null Hypothesis is true.

2. Description

2.1 Definition

• A p-value is a probability associated with your critical value. The critical value depends on the probability you are allowing for a Type I error. It measures the chance of getting results at least as strong as yours if the claim (H0) were true.
• A p-value is a probability that provides a measure of the evidence against the null hypothesis provided by the sample
• Under the assumption that the null hypothesis is true, the p-value is the probability of observing phenomena at least as extreme as that observed.
• The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
• A p-value of less than 0.05 (typically ≤ 0.05) is statistically significant.
• A p-value higher than 0.05 (> 0.05) is not statistically significant and indicates weak evidence against the null hypothesis.

3. Example

$$H_{0}: \mu\geq 70$$

3.1 If p-value = 0.18:

The probability of getting a mean of 68.7 or less from a sample of this size (and variation) is 0.18 or 18%

significance level: $\alpha=0.05$ (1.96 standard deviations, Z-score)

In this case, you can't reject the null hypothesis.

 But if p-value = 0.000001, which is less than the significance level, we can reject the null hypothesis. In short, a small p-value indicates a significant result that has strong evidence to reject the null hypothesis.

4. Reference

https://www.youtube.com/watch?v=5Z9OIYA8He8

https://www.youtube.com/watch?v=KLnGOL_AUgA

https://en.wikipedia.org/wiki/P-value

https://www.mathbootcamps.com/what-is-a-p-value/

https://www.wikihow.com/Calculate-P-Value

https://www.youtube.com/watch?v=eyknGvncKLw&feature=emb_title

https://www.dummies.com/education/math/statistics/how-to-determine-a-p-value-when-testing-a-null-hypothesis/

https://www.researchgate.net/figure/The-normal-distribution-P-values-and-confidence-intervals-A-z-value-of-196-aka-a_fig1_51046126

https://en.wikipedia.org/wiki/Statistical_significance

https://www.simplypsychology.org/p-value.html