Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data.

Key Components:

• Null Hypothesis (\(H_0\)): The default assumption
• Alternative Hypothesis (\(H_a\)): What we’re trying to prove
• Test Statistic: Calculated from sample data
• P-value: Probability of observing the data if \(H_0\) is true
• Significance Level (\(\alpha\)): Threshold for decision (commonly 0.05)

Steps:

1. State the hypotheses (\(H_0\) and \(H_a\))
2. Choose significance level (\(\alpha\))
3. Calculate the test statistic
4. Find the p-value
5. Make a decision: Reject or fail to reject \(H_0\)
6. Interpret results in context

The test statistic for a one-sample t-test is calculated as:

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

Where:

• \(\bar{x}\) = sample mean
• \(\mu_0\) = hypothesized population mean
• \(s\) = sample standard deviation
• \(n\) = sample size

The degrees of freedom are \(df = n - 1\)

Scenario: A professor claims students study an average of 15 hours per week. We sample 30 students to test this claim.

Hypotheses:

\[H_0: \mu = 15 \text{ hours}\] \[H_a: \mu \neq 15 \text{ hours}\]

Significance Level: \(\alpha = 0.05\)

# Generate sample data
set.seed(123)
study_hours <- rnorm(30, mean = 17, sd = 4)

# Perform t-test
t_test_result <- t.test(study_hours, mu = 15)

# Display key results
cat("Sample mean:", round(mean(study_hours), 2), "hours\n")

## Sample mean: 16.81 hours

cat("Test statistic:", round(t_test_result$statistic, 3), "\n")

## Test statistic: 2.529

cat("P-value:", round(t_test_result$p.value, 4))

## P-value: 0.0172

Test Results:

• Test statistic: 2.529
• P-value: 0.0172
• 95% Confidence Interval: [15.35, 18.28]

Decision: Since p-value < 0.05, we reject the null hypothesis.

Conclusion: There is sufficient evidence to conclude that the average study time differs significantly from 15 hours per week. The sample suggests students study approximately 16.8 hours per week.

Key Points:

• Type I Error: False positive (rejecting true \(H_0\))
• Type II Error: False negative (failing to reject false \(H_0\))
• Power = \(1 - \beta\) (probability of correctly rejecting false \(H_0\))