• A special case of the normal distribution with:
  • Mean (\(\mu\)) = 0
  • Standard Deviation (\(\sigma\)) = 1
• Symmetrical and bell-shaped curve
• Frequently used in hypothesis testing and statistical inference

The formula for the standard normal distribution:

\[ f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \]

where: - \(x\) is the standard score (Z-score)

• A continuous probability distribution
• Arises when you sum the squares of independent standard normal variables
• Not symmetric, skewed to the right
• Frequently used in hypothesis tests (e.g., goodness-of-fit, independence)

If \(Z_1, Z_2, ..., Z_k\) are independent standard normal variables, then:

\[ Q = Z_1^2 + Z_2^2 + ... + Z_k^2 \]

follows a Chi-Square distribution with \(k\) degrees of freedom.

• Goodness-of-Fit Test: Check if observed frequencies match expected frequencies
• Test Statistic: \(\chi^2 = \sum \frac{(O - E)^2}{E}\)
  • \(O\): Observed frequency
  • \(E\): Expected frequency
• Compare test statistic to critical value from the Chi-Square distribution

# Example code for Chi-Square test
observed <- c(50, 30, 20)
expected <- c(40, 30, 30)
chisq.test(observed, p = expected/sum(expected))

    Chi-squared test for given probabilities

data:  observed
X-squared = 5.8333, df = 2, p-value = 0.05411