Some Parameters And Their Estimators And Standard Errors

[table class = "table-bordered"] PARAMETER ($\Theta$), ESTIMATOR ($\hat{\Theta}$), STD ERR ($\sigma_{\hat{\Theta}})$, ESTIMATE OF STD ERR ($s_{\hat{\Theta}}$)

$\mu$, $\hat{y}$, $\frac{\sigma}{\sqrt{n}}$, $\frac{s}{\sqrt{n}}$

$\mu_1 - \mu_2$, $\hat{y_1} - \hat{y_2}$, $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$,"$\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2} {n_2}},  n_1 \ge 30,  n_2 \ge 30$"

$\mu_1 - \mu_2$, $\hat{y_1} - \hat{y_2}$, $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$,"$\sqrt{s_p^{2*}(\frac{1}{n_1} + \frac{1}{n_2})},  n_1 < 30  or  n_2 < 30$"

$\frac{\sigma_1^2}{\sigma_2^2}$,$\frac{s_1^1}{s_2^2}$," " ," " , [/table]

*$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$

Confidence Intervals for a Population Parameter $\Theta$ and Test Statistics for $H_0: \Theta = \Theta_0$, where    $\Theta = \mu$  or  $(\mu_1 - \mu_2)$:

[table class = "table-bordered"] SAMPLE SIZE, CONFIDENCE INTERVAL, TEST STATISTIC

Large, $\hat{\Theta} \pm z_{\alpha/2}s_{\hat{\Theta}}$, $z = \frac{\hat{\Theta} - \Theta_0}{s_{\hat{\Theta}}}$

Small,$\hat{\Theta} \pm t_{\alpha/2}s_{\hat{\Theta}}$,$t = \frac{\hat{\Theta} - \Theta_0}{s_{\hat{\Theta}}}$ [/table]

The test statistic for testing the null hypothesis $(H_0: \frac{\sigma_1^2}{\sigma_2^2} = 1)$ is $F = \frac{s_1^2}{s_2^2}$