Hypothesis testing is a statistical method used to make decisions.

We test:

\[H_0: \mu = \mu_0\] \[H_a: \mu \ne \mu_0\]

We calculate:

\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\]

• If \(|t| > t^*\) → Reject \(H_0\)
• If \(|t| \le t^*\) → Fail to reject \(H_0\)

The significance level (\(\alpha\)) is usually 0.05 or 0.01.

The p-value measures how strong the evidence is against \(H_0\).

• Small p-value → strong evidence
• Large p-value → weak evidence

If p-value < \(\alpha\), reject \(H_0\).

Suppose we test:

• \(H_0: \mu = 5\)
• Sample mean = 5.2
  
• Standard deviation = 1.5
  
• Sample size = 30

We calculate the test statistic and make a decision.

set.seed(1)
data <- data.frame(x = rnorm(100))

ggplot(data, aes(x)) +
  geom_histogram(bins = 20)

ggplot(data, aes(sample = x)) +
  stat_qq() +
  stat_qq_line()

plot_ly(x = ~data$x, type = "histogram")

mean(data$x)

## [1] 0.1088874

sd(data$x)

## [1] 0.8981994

Hypothesis testing helps us make decisions using data.

• Start with hypotheses
• Calculate a test statistic
• Compare to a critical value or p-value
• Make a conclusion