Testing of Hypothesis¶
\(\alpha\)
- level of significance
- size of critical region
Confidence level = \((1-\alpha) \times 100 \%\)
The entire distribution is divided into 2 regions
- Critical Region Region of rejection of \(H_0\) it is decided based on \(H_1\)
- Acceptance Region Region of acceptance of \(H_0\)
Population Mean¶
\[ \begin{aligned} H_0: \mu &= \mu_0 & &\text{(Null Hypothesis)} \\ H_1: \mu &< \mu_0, \mu \ne \mu_0, \mu > \mu_0 & &\text{(Alternative Hypothesis)} \\ \end{aligned} \]
| \(\sigma^2\) | \(n\) | Test Statistic/Probability Distribution |
|---|---|---|
| known | any | \(z_c = \frac{\bar x - \mu_0}{\sigma/\sqrt n}\) |
| unknown | \(>30\) | \(z_c = \frac{\bar x - \mu_0}{s/ \sqrt n}\) |
| unknown | \(\le 30\) | \(t_c = \frac{\bar x - \mu_0}{s / \sqrt n}\) |
Critical Region¶
| Left-Tailed | Two-Tailed | Right-Tailed | |
|---|---|---|---|
| \(H_1\) | \(\mu < \mu_0\) | \(\mu \ne \mu_0\) | \(\mu > \mu_0\) |
| p-value | \(F(z_c)\) \(\alpha(t-\text{dist})\) | \(2[ F(-z_c) ]\) \(2 \alpha(t-\text{dist})\) | \(F(-z_c)\) \(\alpha(t-\text{dist})\) |
| Cases | Accept \(H_1\) if \(\begin{aligned} z_c & \le -z_\alpha \\ t_c &\le -t_{(n-1), \alpha} \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) | Accept \(H_1\) if \(\begin{aligned} z_c \le -z_{\alpha/2} &\text{ or } z_c \ge +z_{\alpha/2}\\ t_c \le -t_{(n-1), (\alpha/2)} &\text{ or } t_c \ge +t_{(n-1), (\alpha/2)} \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) | Accept \(H_1\) if \(\begin{aligned} z_c &\ge +z_\alpha \\ t_c &\ge +t_{(n-1), \alpha} \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) |
Proportion¶
\[ \begin{aligned} H_0: p &= p_0 & &\text{(Null Hypothesis)} \\ H_1: p &< p_0, p \ne p_0, p > p_0 & &\text{(Alternative Hypothesis)} \\ z_c &= \frac{\hat p - p_0}{ \sqrt{ \frac{p_0(1-p_0)}{n} } } & & \hat p = \frac x n = \text{Estimated value of } p\\ \end{aligned} \]
Critical Region¶
| Left-Tailed | Two-Tailed | Right-Tailed | |
|---|---|---|---|
| \(H_1\) | \(p < p_0\) | \(p \ne p_0\) | \(p > p_0\) |
| p-value | \(F(z_c)\) | \(2[ F(-z_c) ]\) | \(F(-z_c)\) |
| Cases | Accept \(H_1\) if \(\begin{aligned}z_c &\le -z_\alpha \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) | Accept \(H_1\) if \(\begin{aligned} z_c \le -z_{\alpha/2} &\text{ or } z_c \ge +z_{\alpha/2} \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) | Accept \(H_1\) if \(\begin{aligned} z_c &\ge +z_\alpha \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) |
Variance/SD¶
\[ \begin{aligned} H_0: \sigma^2 &= \sigma^2_0 & &\text{(Null Hypothesis)} \\ H_1: \sigma^2 &< \sigma^2_0, \sigma^2 \ne \sigma^2_0, \sigma^2 > \sigma^2_0 & &\text{(Alternative Hypothesis)} \\ \chi_c^2 &= (n-1) \frac{s^2}{\sigma_0^2} \end{aligned} \]
Critical Region¶
| Left-Tailed | Two-Tailed | Right-Tailed | |
|---|---|---|---|
| \(H_1\) | \(p < p_0\) | \(p \ne p_0\) | \(p > p_0\) |
| p-value | 1 - \(\alpha\)(table) | 1 - \(\alpha\)(table) | 1 - \(\alpha\)(table) |
| Cases | Accept \(H_1\) if \(\begin{aligned}\chi_c^2 &\le \chi^2_{(n-1), (1-\alpha)} \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) | Accept \(H_1\) if \(\begin{aligned}\chi_c^2 \le \chi^2_{(n-1), (1-\alpha/2)} &\text{ or } \chi_c^2 \ge \chi^2_{(n-1), (\alpha/2)} \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) | Accept \(H_1\) if \(\begin{aligned}\chi_c^2 &\ge \chi^2_{(n-1), \alpha} \\ p &\le \alpha \end{aligned}\) else accept \(H_0\) |
Errors¶
| \(H_0\) is true | \(H_0\) is false | \(H_0\) is incorrect | |
|---|---|---|---|
| Reject \(H_0\) | Type 1 Error = \(\alpha\) | Correct | Type 3 Error Right answer to the wrong question |
| Accept \(H_0\) | Correct | Type 2 Error = \(\beta\) |
Type 1 error is alright, but Type 2 error is dangerous
- \(\alpha\) = P(reject \(H_0\) | \(H_0\) is true)
- \(\beta\) = P(accept \(H_0\) | \(H_0\) is false)
Power of Test¶
\[ \text{Power of Test} = 1 - \beta \]
Greater the power of test, the better means that we can more accurately detect when \(H_0\) is false