• A statistical method used to make inferences about a population parameter based on sample data
• Involves formulating two competing hypotheses: null hypothesis (H₀) and alternative hypothesis (H₁)
• Widely used in scientific research, business analytics, and quality control

1. Null Hypothesis (H₀)
2. Alternative Hypothesis (H₁)
3. Test Statistic
4. p-value
5. Significance Level (α)

Let’s consider a simple example:

H₀: μ = 100 (The population mean is equal to 100) H₁: μ ≠ 100 (The population mean is not equal to 100)

• Commonly used significance levels: 0.05, 0.01, 0.001
• For our example, let’s use α = 0.05

The formula for the z-test statistic is:

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

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

# Simulating data
set.seed(123)
sample_data <- rnorm(100, mean = 102, sd = 10)

# Calculating z-statistic
z_stat <- (mean(sample_data) - 100) / 
  (sd(sample_data) / sqrt(length(sample_data)))

# Calculating p-value
p_value <- 2 * (1 - pnorm(abs(z_stat)))

cat("Z-statistic:", round(z_stat, 4), "\n")

## Z-statistic: 3.1814

cat("P-value:", round(p_value, 4))

## P-value: 0.0015

• If p-value < α, reject H₀
• If p-value ≥ α, fail to reject H₀

In our example: - p-value (0.0469) < α (0.05) - Therefore, we reject the null hypothesis

The power of a statistical test is the probability of correctly rejecting a false null hypothesis.

\[ \text{Power} = 1 - \beta \]

Where β is the probability of a Type II error.

• A common application of hypothesis testing
• Used to compare two versions of a webpage, email, or ad
• Helps make data-driven decisions in marketing strategies

Let’s simulate an A/B test for email marketing campaigns:

# Simulating click-through rates for two email versions
set.seed(456)
version_a <- rbinom(1000, 1, 0.05)
version_b <- rbinom(1000, 1, 0.07)

# Performing chi-square test
chi_test <- chisq.test(table(c(version_a, version_b), 
                             c(rep("A", 1000), rep("B", 1000))))

print(chi_test)

## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  table(c(version_a, version_b), c(rep("A", 1000), rep("B", 1000)))
## X-squared = 0.21823, df = 1, p-value = 0.6404

1. Hypothesis testing is a crucial tool in statistical inference
2. The process involves formulating hypotheses, choosing a significance level, calculating test statistics, and making decisions based on p-values
3. Understanding Type I and Type II errors is essential for interpreting results
4. Real-world applications like A/B testing demonstrate the practical value of hypothesis testing in decision-making processes

• Explore more complex hypothesis testing scenarios (e.g., ANOVA, regression analysis)
• Investigate Bayesian approaches to hypothesis testing
• Examine the role of effect size in conjunction with p-values for more comprehensive analysis