Normal Population (Any Sample Size, $n$) $Z = \frac{\bar{X} - \mu_0}{\sqrt{\frac{\sigma^2}{n}}} \sim N(0, 1)$ Non-Normal Population (Large Sample Size, $n > 30$) Population Variance ($\sigma^2$) Known $Z = \frac{\bar{X} - \mu_0}{\sqrt{\frac{\sigma^2}{n}}} \sim N(0, 1) \text{ approx. by CLT}$ Population Variance ($\sigma^2$) Unknown $Z = \frac{\bar{X} - \mu_0}{\sqrt{\frac{s^2}{n}}} \sim N(0, 1) \text{ approx. by CLT}$ Note: We do not need to assume that the population $X$ follows a normal distribution since sample size is large. STEPS TO NOTE Write down $H_0$ and $H_1$. Decide on the significant level. Decide the test statistic. Find the distribution of this statistic assuming that $H_0$ is true. Calculate the test statistic form the sample. Decide whether the test statistic is sufficiently unlikely. Method I : Using rejection region (Classical Method) Method II : Using $p$-value Conclude and interpret the outcome of the test in the context of the question. LEVEL OF SIGNIFICANCE $\alpha %$ level of significance means that there is $\alpha %$ chance of rejecting $H_0$ when $H_0$ is in fact true. Level of Significance $= \mathbf{P}(\text{Reject}\hspace{0.5em} H_0 \mid H_0 \text{ is true})$ P-VALUE The smallest level of significance at which the null hypothesis $H_0$ will be rejected is the $p$-value itself. The smaller the $p$-value is, the stronger evidence against $H_0$. Test Statistics, $Z$ $p$-value $H_1 : \mu > \mu_0$ $\mathrm{P}(Z \geq z_{\text{test}})$ $H_1 : \mu < \mu_0$ $\mathrm{P}(Z \leq z_{\text{test}})$ $H_1 : \mu \neq \mu_0$ $2\mathrm{P}(Z \geq |z_{\text{test}}|)$ Note: $z_{\text{stat}}$, $z_{\text{test}}$ and $z_{\text{calc}}$ are the same. (different textbooks use different notations.)

Hypothesis Testing

Normal Population (Any Sample Size, $n$)

$Z = \frac{\bar{X} - \mu_0}{\sqrt{\frac{\sigma^2}{n}}} \sim N(0, 1)$

Non-Normal Population (Large Sample Size, $n > 30$)

Population Variance ($\sigma^2$) Known

$Z = \frac{\bar{X} - \mu_0}{\sqrt{\frac{\sigma^2}{n}}} \sim N(0, 1) \text{ approx. by CLT}$

Population Variance ($\sigma^2$) Unknown

$Z = \frac{\bar{X} - \mu_0}{\sqrt{\frac{s^2}{n}}} \sim N(0, 1) \text{ approx. by CLT}$

Note: We do not need to assume that the population $X$ follows a normal distribution since sample size is large.

STEPS TO NOTE

Write down $H_0$ and $H_1$.
Decide on the significant level.
Decide the test statistic.
Find the distribution of this statistic assuming that $H_0$ is true.
Calculate the test statistic form the sample.
Decide whether the test statistic is sufficiently unlikely.
Method I : Using rejection region (Classical Method)
Method II : Using $p$-value

Conclude and interpret the outcome of the test in the context of the question.

LEVEL OF SIGNIFICANCE
$\alpha %$ level of significance means that there is $\alpha %$ chance of rejecting $H_0$ when $H_0$ is in fact true.
Level of Significance $= \mathbf{P}(\text{Reject}\hspace{0.5em} H_0 \mid H_0 \text{ is true})$

P-VALUE
The smallest level of significance at which the null hypothesis $H_0$ will be rejected is the $p$-value itself.
The smaller the $p$-value is, the stronger evidence against $H_0$.

Test Statistics, $Z$	$p$-value
$H_1 : \mu > \mu_0$	$\mathrm{P}(Z \geq z_{\text{test}})$
$H_1 : \mu < \mu_0$	$\mathrm{P}(Z \leq z_{\text{test}})$
$H_1 : \mu \neq \mu_0$	$2\mathrm{P}(Z \geq \|z_{\text{test}}\|)$

Note: $z_{\text{stat}}$, $z_{\text{test}}$ and $z_{\text{calc}}$ are the same. (different textbooks use different notations.)

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Test Statistics, \(Z\)	\(p\)-value
\(H_1 : \mu > \mu_0\)	\(\mathrm{P}(Z \geq z_{\text{test}})\)
\(H_1 : \mu < \mu_0\)	\(\mathrm{P}(Z \leq z_{\text{test}})\)
\(H_1 : \mu \neq \mu_0\)	\(2\mathrm{P}(Z \geq \|z_{\text{test}}\|)\)