Central limit theorem and hypothesis tests

Needs independent observations with a common mean $\mu$ and finite variance ${\sigma }^2$; no normality is needed. SE (standard error) is the standard deviation of $\bar{X}$. When $\sigma$ is unknown it is estimated by $s/\sqrt{n}$, where $s$ is the sample standard deviation (divisor $n-1$, below).

Dividing by $n-1$ makes $S^2$ an unbiased estimator of ${\sigma }^2$. Dividing by $n$ instead gives an estimator whose expectation is $\frac{n-1}{n}{\sigma }^2$. Throughout these notes $s^2$ is the $n-1$ version.

For i.i.d. observations with finite mean and variance, $\bar{X}$ is approximately $N(\mu ,{\sigma }^2/n)$ for large $n$, whatever the shape of the population. Equivalently, the sum $S_n=X_1+\cdots +X_n$ is approximately $N(n\mu ,n{\sigma }^2)$. If the population is itself normal, $\bar{X}$ is exactly normal for every $n$.

For $X\sim \text{Binomial}(n,p)$, which has mean $np$ and variance $np(1-p)$. A common rule of thumb is $np\ge 5$ and $n(1-p)\ge 5$ (some use 10). $\Phi$ is the standard normal CDF. The $\pm 0.5$ moves each integer boundary to the middle of the gap between integers.

Independent observations; population normal, or $n$ large (CLT). Commonly used values: $z_{0.05}=1.645$ (90%), $z_{0.025}=1.96$ (95%), $z_{0.005}=2.576$ (99%).

Population normal (for large $n$ the interval is approximately valid without normality, and $t\approx z$). Here $s^2$ is the sample variance with divisor $n-1$. The t distribution has $n-1$ degrees of freedom and heavier tails than the normal, so $t_{\alpha /2,n-1}>z_{\alpha /2}$; it approaches $z_{\alpha /2}$ as $n$ grows.

Large-sample (normal) interval, valid when $n\hat{p}$ and $n(1-\hat{p})$ are both large. $\hat{p}=x/n$ is the sample proportion.

$E$ is the required margin of error (half-width). Round $n$ up to the next integer.

$\beta$ depends on which alternative value is true. For a composite $H_0$ (such as $\mu \le {\mu }_0$), $\alpha$ is the largest Type I error probability over $H_0$. For a fixed sample size and a fixed alternative, reducing $\alpha$ increases $\beta$. Increasing $n$ can reduce both.

For a one-sided test use $1-\Phi (z)$ when $H_1:\mu >{\mu }_0$ (upper tail) and $\Phi (z)$ when $H_1:\mu <{\mu }_0$ (lower tail).

Use when $\sigma$ is known and the population is normal or $n$ is large. Under $H_0:\mu ={\mu }_0$, $z\sim N(0,1)$ (exactly for normal data, approximately otherwise).

Use when $\sigma$ is unknown and the population is normal; $s$ uses divisor $n-1$. Under $H_0$, $t$ follows a t distribution with $n-1$ degrees of freedom.

Symmetric about 0 with heavier tails than $N(0,1)$; it tends to $N(0,1)$ as $\nu \to \infty$. For normal data, $\frac{\bar{X}-\mu }{S/\sqrt{n}}$ has this distribution with $\nu =n-1$.

Independent samples from normal populations assumed to share the same variance; tests $H_0:{\mu }_1={\mu }_2$. $s_p^2$ is the pooled sample variance.

For $n$ matched pairs (for example before/after on the same subjects), take differences $d_i=x_i-y_i$. $\bar{d}$ and $s_d$ are their mean and sample SD (divisor $n-1$). The differences are assumed normal; tests $H_0:{\mu }_d=0$. Do not use the two-sample formula on paired data.

Independent samples, variances ${\sigma }_1^2,{\sigma }_2^2$ known, populations normal or both samples large; tests $H_0:{\mu }_1={\mu }_2$.

Large $n$. Note that the standard error uses $p_0$ (the value under $H_0$), whereas the confidence interval uses $\hat{p}$.

Independent large samples; tests $H_0:p_1=p_2$. Under $H_0$ the common proportion is estimated by the pooled $\hat{p}$.

Takes only non-negative values and is right-skewed. The sum of independent chi-squared variables is chi-squared, with the degrees of freedom added.

$O$ = observed count, $E$ = expected count under $H_0$, $N$ = total count. Large values mean poor agreement, so these two tests are upper-tailed: reject when ${\chi }^2$ exceeds ${\chi }_{\alpha ,\text{df}}^2$. The chi-squared approximation needs expected counts that are not tiny (a common rule is $E\ge 5$).

$k$ categories, $m$ parameters estimated from the data. If $H_0$ fully specifies the probabilities, $m=0$ and $E_i=n{p}_i$.

$R_i$ = row total, $C_j$ = column total, $N$ = grand total. $H_0$: the two classifications are independent.

Tests $H_0:{\sigma }^2={\sigma }_0^2$ for a normal population ($s^2$ with divisor $n-1$); the result is sensitive to non-normality. Unlike the count tests above, this test can be two-sided: for $H_1:{\sigma }^2\ne {\sigma }_0^2$ reject if ${\chi }^2>{\chi }_{\alpha /2,n-1}^2$ or ${\chi }^2<{\chi }_{1-\alpha /2,n-1}^2$. For $H_1:{\sigma }^2>{\sigma }_0^2$ reject if ${\chi }^2>{\chi }_{\alpha ,n-1}^2$.

With the upper-tail convention, ${\chi }_{\alpha /2,n-1}^2$ is the larger critical value, so it gives the lower limit. The interval is not symmetric about $s^2$.