• The p-value helps to determine the significance of your results in hypothesis testing.
• It is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
• Key Idea: A small p-value suggests that the null hypothesis may not hold.

• The p-value is computed as:

\[ P = P(T \geq t | H_0) \]

where: - \(t\) is the observed test statistic - \(H_0\) is the null hypothesis

• We are testing if the mean of a sample is equal to a given value \(\mu_0\).
• Null Hypothesis \(H_0: \mu = \mu_0\)
• Alternative Hypothesis \(H_1: \mu \neq \mu_0\)

set.seed(123)

sample_data <- rnorm(30, mean = 5, sd = 2)

t_test_result <- t.test(sample_data, mu = 5)

t_test_result$p.value

## [1] 0.7944204

• The p-value provides an essential tool for hypothesis testing.
• A p-value below a significance level (e.g., 0.05) suggests rejecting the null hypothesis.
• Visualizing the p-value can give us more intuition about the hypothesis test.