Hypothesis testing is a method of making statistical decisions using experimental data. It is used to infer whether a hypothesis about a population parameter should be accepted or rejected.

1. State the null hypothesis \(H_0\) and alternative hypothesis \(H_a\).
2. Choose a significance level \(\alpha\).
3. Collect data and calculate a test statistic.
4. Make a decision: reject or fail to reject \(H_0\).

Example:

• Null hypothesis \(H_0: \mu = 50\)
• Alternative hypothesis \(H_a: \mu \neq 50\)

Two-tailed test with significance level \(\alpha = 0.05\).

Test statistic (z-test for mean):

\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

Where: - \(\bar{x}\): sample mean - \(\mu_0\): hypothesized mean - \(\sigma\): population standard deviation - \(n\): sample size

set.seed(123)
sample_data <- rnorm(100, mean = 51, sd = 10)

sample_mean <- mean(sample_data)
sample_sd <- sd(sample_data)
n <- length(sample_data)
z_stat <- (sample_mean - 50) / (sample_sd / sqrt(n))
p_value <- 2 * (1 - pnorm(abs(z_stat)))

list(z_stat = round(z_stat, 2), p_value = round(p_value, 4))

## $z_stat
## [1] 2.09
## 
## $p_value
## [1] 0.037

• Hypothesis testing helps make data-driven decisions.
• Small p-value \((p < \alpha)\) means strong evidence against \(H_0\).
• Large sample size improves the reliability of results.
• Visualization enhances understanding of critical regions and p-values.

Thank you for learning about Hypothesis Testing.