Demonstration: Chi-Square Test for Population Variance in R

In this section, we demonstrate how to perform a hypothesis test for the population variance using the Chi-Square distribution.

We use the example from the blog post, where we test whether the population variance of metal rod lengths is greater than \(\sigma_0^2 = 0.04\) based on a sample variance of \(s^2 = 0.10\) from \(n = 10\) samples, at a significance level of \(\alpha = 0.05\).

The code below: 1. Computes the Chi-Square test statistic.
2. Determines the critical value and p-value for the right-tailed test.
3. Visualizes the Chi-Square distribution with the rejection region shaded and the test statistic marked.
4. Provides a conclusion based on the comparison between the test statistic and critical value.

# --- Chi-Square Test for Population Variance ---

# Given data
n <- 10                  # sample size
s2 <- 0.10               # sample variance
sigma0_sq <- 0.04        # hypothesized population variance
alpha <- 0.05            # significance level

# Compute the test statistic
chi_sq <- (n - 1) * s2 / sigma0_sq
df <- n - 1

# Compute the critical value for a right-tailed test
chi_crit <- qchisq(1 - alpha, df)

# Compute the p-value
p_value <- 1 - pchisq(chi_sq, df)

# Print results
cat("Chi-Square Test for Population Variance\n")
## Chi-Square Test for Population Variance
cat("----------------------------------------\n")
## ----------------------------------------
cat(sprintf("Sample size (n): %d\n", n))
## Sample size (n): 10
cat(sprintf("Sample variance (s^2): %.3f\n", s2))
## Sample variance (s^2): 0.100
cat(sprintf("Hypothesized variance (σ0^2): %.3f\n", sigma0_sq))
## Hypothesized variance (σ0^2): 0.040
cat(sprintf("Test statistic (χ²): %.3f\n", chi_sq))
## Test statistic (χ²): 22.500
cat(sprintf("Critical value (χ²_crit, α=%.2f): %.3f\n", alpha, chi_crit))
## Critical value (χ²_crit, α=0.05): 16.919
cat(sprintf("p-value: %.4f\n", p_value))
## p-value: 0.0074
# Decision rule
if (chi_sq > chi_crit) {
  cat("\nDecision: Reject H0 (variance is significantly greater than hypothesized)\n")
} else {
  cat("\nDecision: Fail to reject H0 (no evidence variance is greater than hypothesized)\n")
}
## 
## Decision: Reject H0 (variance is significantly greater than hypothesized)
# --- Visualization ---
# Plot Chi-Square distribution with rejection region and test statistic
curve(dchisq(x, df), from = 0, to = 30, 
      lwd = 2, col = "darkblue", 
      ylab = "Density", xlab = expression(chi^2), 
      main = bquote("Chi-Square Distribution (df = " ~ .(df) ~ ")"))

# Shade rejection region
x_shade <- seq(chi_crit, 30, length.out = 200)
y_shade <- dchisq(x_shade, df)
polygon(c(chi_crit, x_shade, 30), c(0, y_shade, 0), 
        col = rgb(1, 0, 0, 0.3), border = NA)

# Add vertical line for test statistic
abline(v = chi_sq, col = "darkgreen", lwd = 2, lty = 2)

# Annotate key points
text(chi_sq + 0.8, 0.02, labels = bquote(chi^2[obs] == .(round(chi_sq, 2))), col = "darkgreen", pos = 4)
text(chi_crit - 0.8, 0.02, labels = bquote(chi^2[crit] == .(round(chi_crit, 2))), col = "red", pos = 2)

legend("topright", legend = c("Chi-Square PDF", "Rejection Region", "Observed χ²"),
       col = c("darkblue", rgb(1,0,0,0.4), "darkgreen"),
       lty = c(1, NA, 2), lwd = c(2, NA, 2), pch = c(NA, 15, NA), pt.cex = 1.5)