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https://www.youtube.com/watch?v=N5cbASDXl1w&list=PLsri7w6p16vvQCo9pmuRNY_SYoOGB6bWM&index=3
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Case1: "enough sample", "variance of 2 populations is known"
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$$$\mu_A$$$, $$$\mu_B$$$: mean of populations
$$$\sigma_A^2$$$, $$$\sigma_B^2$$$: variance of 2 populations
$$$\mu_A - \mu_B = \delta$$$
* Difference between mean of A and mean of B
$$$\bar{x}_A - \bar{x}_B = \delta$$$
* Difference between mean of A sample and mean of B sample
* predicted value for $$$\mu_A - \mu_B = \delta$$$
* You can normalize it
$$$z = \dfrac{(\bar{x}_A-\bar{x}_B) - (\mu_A - \mu_B)}{\sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}}$$$
* Then, z follows $$$\mathcal{N}(mu=0,std=1)$$$
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* Let's calculate trusted region
$$$100(1-\alpha)\%$$$ trusted region for $$$\mu_A-\mu_B$$$
* If $$$\alpha=0.05$$$, it's $$$95\%$$$
* If it's 2 sided test ($$$\frac{\alpha}{2}$$$), let's calculate trusted region
$$$(\bar{x}_A - \bar{x}_B) - z_{\frac{\alpha}{2}} \sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}
\le \mu_A-\mu_B
\le (\bar{x}_A - \bar{x}_B) + z_{\frac{\alpha}{2}} \sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}$$$
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Meaning
* What you want to find is $$$\mu_A-\mu_B$$$
* When you calculate $$$\mu_A-\mu_B$$$,
you have under bound $$$(\bar{x}_A - \bar{x}_B) - z_{\frac{\alpha}{2}} \sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}$$$ and upper bound $$$(\bar{x}_A - \bar{x}_B) + z_{\frac{\alpha}{2}} \sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}$$$
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Let's create hypothesis
$$$H_0: \mu_A - \mu_B = 0$$$
$$$H_1: \mu_A - \mu_B \ne 0$$$
* Left sided test
$$$H_0: \mu_A - \mu_B = 0$$$
$$$H_1: \mu_A - \mu_B \lt 0$$$
* Right sided test
$$$H_0: \mu_A - \mu_B = 0$$$
$$$H_1: \mu_A - \mu_B \gt 0$$$
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* Test statistics z for testing $$$H_0: \mu_A - \mu_B = 0$$$
$$$z = \dfrac{(\bar{x}_A-\bar{x}_B) - (\mu_A - \mu_B)}{\sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}}$$$
Since $$$H_0: \mu_A - \mu_B = 0$$$
$$$z = \dfrac{(\bar{x}_A-\bar{x}_B)}{\sqrt{\frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}}}$$$
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Case2:
* "enough sample", "variance of 2 populations is unknown"
* You don't know $$$\sigma_A^2, \sigma_B^2$$$ of 2 populations
* You can predict them via samples
$$$s_A^2, s_B^2$$$
* Different between mean of samples
$$$\bar{x}_A - \bar{x}_B \\
= E(\bar{x}_A - \bar{x}_B) \\
= E(\bar{x}_A) - E(\bar{x}_B)) \\
=\mu_A-\mu_B$$$
* Different between variance of samples
$$$s^2(\bar{x}_A - \bar{x}_B) \\
= s^2(\bar{x}_A) + s^2(\bar{x}_B)) - 2\times Cov(\bar{x}_A),\bar{x}_B)) \\
= \frac{S_{x_A}^2}{n_A} + \frac{S_{x_B}^2}{n_B}$$$
precondition: $$$n_A,n_B$$$ should $$$\ge$$$ 30
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Trusted region wrt $$$\mu_A - \mu_B$$$ by $$$100(1-\alpha)\%$$$
$$$(\bar{x}_A - \bar{x}_B) - z_{\frac{\alpha}{2}} \times \sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}
\le \mu_A-\mu_B
\le (\bar{x}_A - \bar{x}_B) + z_{\frac{\alpha}{2}} \times \sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}$$$
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2 sided test
$$$H_0: \mu_A-\mu_B=0$$$
$$$H_1: \mu_A-\mu_B\ne 0$$$
left sided test
$$$H_0: \mu_A-\mu_B=0$$$
$$$H_1: \mu_A-\mu_B\lt 0$$$
left sided test
$$$H_0: \mu_A-\mu_B=0$$$
$$$H_1: \mu_A-\mu_B\gt 0$$$
test statistics z
$$$z = \dfrac{(\bar{x}_A - \bar{x}_B)-(\mu_A - mu_B)}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}}$$$
since $$$H_0: \mu_A-\mu_B=0$$$
$$$z = \dfrac{(\bar{x}_A - \bar{x}_B)}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}}$$$
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