User:Deep evil/Sandbox

1
$$H_0 : p_1 = p_2$$

$$z = \frac{\hat{p}_1 - \hat{p}_2}{\mathrm{SE}_{Dp}}$$

$$\mathrm{SE}_{Dp} = \sqrt{ \hat{p}(1-\hat{p}) ( \frac{1}{n_1} + \frac{1}{n_2})}$$

$$\hat{p} = \frac{X_1 + X_2}{n_1+n_2}$$

$$\mathrm{RR} = \frac{\hat{p}_1}{\hat{p}_2}$$

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2
$$\mathrm{H}_0 : \mu_t = 0.5714 \mathrm{sec}$$

$$ \mathrm{H}_\alpha : \mu_t > 0.5714 \mathrm{sec} $$

$$ s=0.1429 $$

$$ t = \frac{2.1287-0.5714}{\frac{0.1429}{\sqrt{30}}}$$

$$H_0$$

$$H_0: \mu_{\mathrm{m-t}} = 0$$

$$H_\alpha: \mu_{\mathrm{m-t}} > 0$$

$$ t = 59.31$$

$$a = .01 $$

$$ p = 0^+$$

$$p < a$$

$$d=0.1379$$

$$s=0.2964$$

$$t=\frac{.1379-0}{\frac{0.1964}{\sqrt{30}}}$$

$$t=2.54$$

$$a=0.01$$

$$p=0.0082$$

$$p > a$$

$$H_0: \mu_t = \mu_o$$

$$H_\alpha: \mu_t > \mu_o$$

$$\bar x_t = 2.6237$$

$$\bar x_o = 2.1187$$

$$df=57.9848$$

$$s_t=0.1405$$

$$s_o=0.1428$$

$$t=13.8000$$

$$a=0.01$$

$$p=0^+$$

$$\bar x = 2.12$$

$$\mu_t$$

$$\mu_{m-t}$$

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3
$$\chi^2 = \sum \frac{\mathrm{(observed - expected)}^2} {\mathrm{expected}}$$

$$\chi^2 = \frac{(210-224)^2}{224} + ... +\frac{(530-558)^2}{558}$$

$$\chi^2 = 6.5722$$

$$\mathrm{df} = (4-1)(2-1) = 3$$

$$\mathrm{p-value} = .0374$$

$$\mathrm{P} (\chi^2 > 6.5722) = .0374$$

$$\chi^2 = \frac{(505-432.2182)^2}{432.2182} + ... +\frac{(47-12.6234)^2}{12.6234}$$

$$\chi^2 = 236.0876$$

$$\mathrm{df} = (3-1)(3-1) = 4$$

$$\mathrm{p-value} = 0^+$$

$$\mathrm{P} (\chi^2 > 236.0876) = 0^+$$

4
$$\chi^2\ \mathrm{test\ of\ independence}$$

$$\chi^2 = \sum \frac{\mathrm{(observed - expected)}^2} {\mathrm{expected}}$$

$$\chi^2 = \frac{(20-27.3995)^2}{27.3995} + ... + \frac{(30-36.9054)^2}{36.9054}$$

$$\chi^2 = 6.8519$$

$$\mathrm{p-value} = 0.2319$$

$$df = (6-1)(2-1) = 5$$

$$\alpha = .05$$

$$\chi^2 = \frac{(6-8.1176)^2}{8.1176} + ... + \frac{(7-4.5882)^2}{4.5882}$$

$$\chi^2 = 4.9236$$

$$df = (2-1)(3-1) = 2$$

$$\mathrm{p-value} = .0853$$

$$\chi^2 = \frac{(24-28.682)^2}{28.682} + ... + \frac{(10-14.682)^2}{14.682}$$

$$\chi^2 = 4.4252$$

$$df = (2-1)(2-1) = 1$$

$$\mathrm{p-value} = .0354$$

5
$$\mu_y = \beta_0 + \beta_1x$$

$$(x_1,y_1), (x_2,y_2), ..., (x_n,y_n)$$

$$y_i = \beta_0 + \beta_1 x_i + \in_i$$

$$e_i = \mathrm{observed\ response - predicted\ response}$$

$$ = y_i - \hat{y}_i$$

$$ = y_i - b_0 - b_1x_i$$

$$b_0 \pm t^*\mathrm{SE}_{b_0}$$

$$b_1 \pm t^*\mathrm{SE}_{b_1}$$

$$t= \frac{b_1}{\mathrm{SE}_b}$$

$$\hat{y} \pm t^*\mathrm{SE}_{\hat{y}}$$