ข้อสอบ PAT 1 - กุมภาพันธ์ 2562

$$\begin{gathered} - \mathrm{จากข้อมูลของประชากร} \\ 2, 2+d, 2+2d, 2+3d, ..., 2+30d \\ \mathrm{จะเห็นว่าข้อมูลเรียงกันเป็นลำดับเลขคณิตโดย} N=31 \mathrm{จำนวน} \end{gathered}$$

$$\begin{gathered} - \mathrm{จากสูตรค่าเฉลี่ยเลขคณิต} \mu = \frac{\sum x_i}{N} \\ \mu = \frac{2+\left(2+d\right)+\left(2+2d\right)+\left(2+3d\right)+...+\left(2+30d\right)}{31} \\ = \frac{2\left(31\right)+\left(d+2d+3d+...+30d\right)}{31} \\ \mu = \frac{2\left(31\right)+\left(1+2+3+...+30\right)d}{31} \end{gathered}$$

$$\begin{gathered} - \mathrm{จากสูตร} \sum _{i=1}^{N}i = 1+2+3+...+n = \frac{n\left(n+1\right)}{2} \\ - \mathrm{จากสูตร} \sum _{i=1}^N{i}^2 = 1^2+2^2+3^2+...+n^2 = \frac{n\left(n+1\right)\left(2n+1\right)}{6} \\ \therefore \mu = \frac{2\cancel{\left(31\right)}+\left[{\displaystyle \frac{30\left(31\right)}{2}}\right]d}{\cancel{31}} \end{gathered}$$

$$\begin{gathered} \mu = 2+15d \to \boxed{1} \\ - \mathrm{โจทย์ให้ความแปรปรวนของข้อมูลเท่ากับ} 320 \\ \therefore {\sigma }^2 = 320 \\ \mathrm{จากสูตร} {\sigma }^2 = \frac{{\displaystyle \sum _{i=1}^N}{\left(x_i-\mu \right)}^2}{N} \end{gathered}$$

$$\begin{gathered} 320 = \frac{{\left[2-\left(2+15d\right)\right]}^2+{\left[\left(2+d\right)-\left(2+15d\right)\right]}^2+...+{\left[\left(2+30d\right)-\left(2-15d\right)\right]}^2}{31} \\ = \frac{{\left(-15d\right)}^2+{\left(-14d\right)}^2+...+0+...+{\left(14d\right)}^2+{\left(15d\right)}^2}{31} \end{gathered}$$

$$\begin{gathered} = \frac{d^2\left[{15}^2+{14}^2+...+0+...+{14}^2+{15}^2\right]}{31} \\ = \frac{d^2\left[2\left({15}^2+{14}^2+{13}^2+...+2^2+1^2\right)\right]}{31} \\ = \frac{2d^2\left[{\displaystyle \frac{15\left(16\right)\cancel{\left(31\right)}}{6}}\right]}{\cancel{31}} \end{gathered}$$

$$\begin{gathered} 320 = 2d^2\left[\frac{15\left(16\right)}{6}\right] \\ 160 = d^2\left[\frac{5\left(16\right)}{2}\right] \\ 10 = \frac{5}{2}{d}^2 \end{gathered}$$
$$\begin{gathered} d^2 = 4 \\ d = \pm 2 \\ \therefore d = 2 (d \mathrm{เป็นจำนวนจริง}) \\ \mathrm{ดังนั้น} \mathrm{จาก} \boxed{1} \mathrm{จะได้} \mu = 2+15\left(2\right) = 32 \end{gathered}$$