Hypothesis testing for single means and proportions is a key statistical tool. It involves formulating null and alternative hypotheses, calculating test statistics, and interpreting results using p-values and significance levels.

This process helps determine if sample data provides enough evidence to reject a null hypothesis about a population parameter. Understanding these concepts is crucial for making informed decisions based on statistical analysis in various fields.

Hypothesis Testing for a Single Mean and Proportion

Formulation of hypotheses

• Null hypothesis ($H_0$)
  • States population parameter (mean or proportion) equals specific value
  • Assumed true unless strong evidence against it
  • Single mean: $H_0: \mu = \mu_0$ ($\mu$ population mean, $\mu_0$ hypothesized value)
  • Single proportion: $H_0: p = p_0$ ($p$ population proportion, $p_0$ hypothesized value)
• Alternative hypothesis ($H_a$ or $H_1$)
  • Contradicts null hypothesis
  • One-sided (left-tailed or right-tailed) or two-sided
    • Left-tailed: $H_a: \mu < \mu_0$ or $H_a: p < p_0$
    • Right-tailed: $H_a: \mu > \mu_0$ or $H_a: p > p_0$
    • Two-sided: $H_a: \mu \neq \mu_0$ or $H_a: p \neq p_0$

Calculation of test statistics

• Test statistic
  • Standardized value measuring difference between sample statistic and hypothesized value under null hypothesis
  • Single mean (z-test): $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$
    • $\bar{x}$ sample mean
    • $\mu_0$ hypothesized mean
    • $\sigma$ population standard deviation
    • $n$ sample size
  • Single mean (t-test): $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
    • $s$ sample standard deviation (used when $\sigma$ unknown)
  • Single proportion: $z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0) / n}}$
    • $\hat{p}$ sample proportion
    • $p_0$ hypothesized proportion
• P-value
  • Probability of obtaining sample statistic as extreme as or more extreme than observed, assuming null hypothesis true
  • Calculated using test statistic and appropriate distribution (standard normal for z-tests, t-distribution for t-tests)
  • Left-tailed test: p-value = P(Z < z) or P(T < t)
  • Right-tailed test: p-value = P(Z > z) or P(T > t)
  • Two-sided test: p-value = 2 × min(P(Z < z), P(Z > z)) or 2 × min(P(T < t), P(T > t))
• Critical value
  • Value that separates rejection region from non-rejection region in hypothesis testing

Interpretation of hypothesis tests

• Significance level ($\alpha$)
  • Probability of rejecting null hypothesis when actually true (Type I error)
  • Common values: 0.01, 0.05, 0.10
• Comparing p-value to significance level
  1. If p-value ≤ $\alpha$, reject null hypothesis
     • Conclude sufficient evidence to support alternative hypothesis
  2. If p-value > $\alpha$, fail to reject null hypothesis
     • Conclude not enough evidence to support alternative hypothesis
• Confidence intervals
  • Interval estimate of population parameter providing precision of estimate
  • Single mean (z-interval): $\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$
  • Single mean (t-interval): $\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$
  • Single proportion: $\hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$
  • If confidence interval contains hypothesized value, supports failing to reject null hypothesis; otherwise, supports rejecting null hypothesis
• Degrees of freedom
  • Number of independent observations in a sample that are free to vary, affecting the shape of the t-distribution

Additional Considerations in Hypothesis Testing

• Statistical power: Probability of correctly rejecting a false null hypothesis
• Effect size: Measure of the magnitude of the difference between groups or the strength of a relationship
• Central limit theorem: States that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population distribution