Biostats Midterm II Review

Empirical Rule (68-95-99.7 Rule)

For a normal random variable $X$ with mean $\mu$ and SD $\sigma$: $68\%$ falls within $\mu \pm \sigma$

$95\%$ within $\mu \pm 2\sigma$

$99.7\%$ within $\mu \pm 3\sigma$.

Calculating Normal Probabilities (Left, Right, Between)

Convert to a $Z$-score and find the CDF area:

Left-sided $= \text{CDF value}$

Right-sided $= 1 - \text{CDF value}$

Between $= \text{Area}_{\text{upper}} - \text{Area}_{\text{lower}}$

Calculating Percentiles from a Normal Distribution

1) Find $\text{Area} = \frac{\%}{100}$

2) Look up the corresponding $Z$-score

3) Convert to a raw score using $X = \mu + Z \cdot \sigma$

Central Limit Theorem (CLT) & Sample Size Rules

As sample size $n$ increases (Law of Large Numbers), the sampling distribution of $\bar{X}$ approaches normality. Universal rule: $n \ge 30$ if the population is not normal ($n < 30$ is acceptable if normality is assumed).

Confidence Level & Critical Value Cutoffs

For confidence level $1 - \alpha$:

Lower critical value is at $\frac{\alpha}{2}$, and upper critical value is at $1 - \frac{\alpha}{2}$.

Common $Z$ Critical Values (90%, 95%, 99%)

$$90\% \rightarrow 1.645$$

$$95\% \rightarrow 1.960$$

$$99\% \rightarrow 2.576$$

Interpreting a Confidence Interval for Two Means ($\mu_1 - \mu_2$)

$H_0: \mu_1 - \mu_2 = 0$ (or $\mu_1 = \mu_2$). If $0$ is inside the confidence interval, the two population means may be equal (no significant difference).

Population Variances

$\sigma_1^2=\sigma_2^2$ or $\sigma_1^2\ne\sigma_2^2$

Confidence Interval / Distribution for 1 Variance

Uses the Chi-Square ($\chi^2$) distribution with $\text{df} = n - 1$. Test statistic: $\chi^2 = \frac{(n - 1)s^2}{\sigma^2} \sim \chi^2_{n-1}$.

Confidence Interval for 2 Variances ($F$-Test)

Compares $\sigma_x^2$ and $\sigma_y^2$ as a ratio. If $1$ is inside the confidence interval, the two population variances are considered equal ($\sigma_x^2 = \sigma_y^2$).

Type I Error ($\alpha$) vs. Type II Error ($\beta$)

Type I Error ($\alpha$): Wrongly rejecting a true $H_0$ (false positive).

Type II Error ($\beta$): Wrongly failing to reject a false $H_0$ (false negative).

Null ($H_0$) vs. Alternative ($H_A$) Hypotheses

$H_0$ assumes no effect / chance alone. $H_A$ contradicts $H_{A}$ : Left-sided ($\mu_1 < \mu_2$), Right-sided ($\mu_1 > \mu_2$), or Two-sided ($\mu_1 \ne \mu_2$).

P-value Definition

The probability of obtaining a test result at least as extreme as the observed data, assuming the null hypothesis ($H_0$) is true.

Decision Rule for Hypothesis Testing ($p$ vs. $\alpha$)

If $p < \alpha$: Reject $H_0$

If $p > \alpha$: Fail to reject $H_0$ (do not "accept" $H_0$).

One-Sample $t$-Test (Assumptions & $H_0$)

Tests one population mean ($H_0: \mu = \mu_0$).

Assumptions: data are continuous, randomly sampled, and normally distributed.

Two-Sample $t$-Test: Pooled vs. Welch's

Tests two independent means ($H_0: \mu_x = \mu_y$). Assumptions: independent, random, normal. Use a Pooled $t$-test for equal variances (homogeneity: $\sigma_1^2 = \sigma_2^2$) and Welch's $t$-test for unequal variances (heterogeneity: $\sigma_1^2 \ne \sigma_2^2$).

Paired $t$-Test (Assumptions & Hypotheses)

Used for dependent/pre-post data ($H_0: \mu_d = 0$). Directional alternatives: $\mu_d < 0$ (increasing effect from pre to post), $\mu_d > 0$ (decreasing effect), or $\mu_d \ne 0$ (two-sided effect).

Wilcoxon Signed-Rank Test

Nonparametric test for one-sample or paired medians ($H_0: M = M_0$).

Assumptions: data are continuous and randomly sampled.

Mann-Whitney $U$ (Rank-Sum) Test

Nonparametric test for two independent population medians ($H_0: M_x = M_y$).

Assumptions: independent, random, continuous, and distributions have similar shapes (differ only by location/median).

Shapiro-Wilk Test

Tests whether data deviate from normality (useful for small $n$). $H_0$: Data are normally distributed. If $p > \alpha$, conclude data do not deviate from normality.

$F$-Test for Two Variances

Compares two population variances to check for significant differences. Null hypothesis: $H_0: \frac{\sigma_1^2}{\sigma_2^2} = 1$.

$Z$-Test for One and Two Proportions

One-proportion: tests $H_0: p = p_0$. Two-proportion: tests $H_0: p_1 = p_2$.

Chi-Squared ($\chi^2$) Test for $K$ Proportions

Tests $H_0: p_1 = p_2 = \dots = p_k$. Rejects $H_0$ when observed counts ($O$) differ greatly from expected counts ($E$), driving $\chi_{\text{obs}}^2$ to infinity.

Assumptions: all $E \ge 5$ (never $E < 1$; $1 \le E < 5$ is marginally acceptable).

Statistical Power (Definition & Standard Threshold)

$\text{Power} = 1 - \beta$: The probability that a test will correctly reject a false $H_0$ (detect an effect when one truly exists).

Standard Min. for Power

Standard minimum threshold is $80\%$ ($0.80$).

Power Analysis Parameters for a $t$-Test

Four interrelated components (solve for $n$): $\text{Power}$ (typically $80\%$), Significance level $\alpha$ (typically $5\%$), Sample size ($n$), and Effect size (Cohen's $d$).

Cohen's $d$ Formula & Benchmarks

Standardized mean difference: $d = \frac{\mu_A - \mu_0}{\sigma} = \frac{\delta}{\sigma}$.

Cohen’s d Benchmarks

$< 0.2$ = negligible

$0.2$ = small

$0.5$ = medium

$> 0.8$ = large.