https://www.youtube.com/watch?v=lQx0mYMQ5C8&list=PLsri7w6p16vuFEiIVqKFwbB8MlHesFlEM&index=2&t=0s ================================================================================ ================================================================================ Weight in kg Before taking drug After taking drug $$$H_0$$$:no weight change --meaning--> $$$\bar{x}_{\text{before}}-\bar{x}_{\text{after}}=0$$$ $$$H_1$$$:weight change --meaning--> $$$\bar{x}_{\text{before}}-\bar{x}_{\text{after}}\ne 0$$$ ================================================================================ $$$\bar{x}_1$$$: before drug $$$\bar{x}_2$$$: after drug $$$n$$$: number of sample $$$x_1-x_2=d_x$$$ Before and after at each period $$$\bar{d}_x = \frac{\sum\limits (x_1-x_2)}{n}$$$ Before and after for entire period $$$S_{d_x}^2 = \dfrac{\sum\limits(d_x-\bar{d}_x)^2}{n-1}$$$ Variance $$$S_{d_x} = \sqrt{S_{d_x}^2} = \sqrt{\dfrac{\sum\limits(d_x-\bar{d}_x)^2}{n-1}}$$$ Std $$$s_{\bar{d}_x} = \frac{S_{d_x}}{\sqrt{n}}$$$ ================================================================================ Let's calculate $$$\bar{d}_x = \dfrac{(75-73)+(74-74)+\cdots+(68-67)}{30} = 3.90$$$ $$$S_{d_x}^2 = \dfrac{(2-3.9)^2+(0-3.9)^2+\cdots+(1-3.9)^2}{30-1} = 13.197$$$ $$$S_{d_x} = \sqrt{13.197} = 3.633$$$ $$$s_{\bar{d}_x} = \frac{3.633}{\sqrt{30}}$$$ ================================================================================ test hypothesis p value=0.05 degree of freedom = 29 $$$t_{(29,0.025)} = 2.045$$$ $$$d_x-t_{(n-1,\frac{\alpha}{2})} \times \frac{S_{d_x}}{\sqrt{n}} \le t \le d_x+t_{(n-1,\frac{\alpha}{2})} \times \frac{S_{d_x}}{\sqrt{n}}$$$ $$$2.9 - 2.045 \times \frac{3.633}{\sqrt{30}} \le t \le 2.9 + 2.045 \times \frac{3.633}{\sqrt{30}}$$$ $$$2.544 \le t \le 5.256$$$ $$$t = \dfrac{d_x}{\frac{s_{d_x}}{\sqrt{n}}} = \frac{3.9}{0.663} = 5.880$$$ Red areas: rejection area $$$H_0$$$ is rejected