the table below shows the probability distribution of a random variable z. \n|z|p(z)|\n|6|0.2|\n|7|0.4|\n|8|0…

Explanation:

Step1: Calculate the expected value $E(Z)$

$E(Z)=\sum_{i}z_iP(z_i)=6\times0.2 + 7\times0.4+8\times0 + 9\times0.4=6\times0.2+2.8 + 0+3.6=1.2 + 2.8+3.6 = 7.6$

Step2: Calculate the variance $Var(Z)$

$Var(Z)=\sum_{i}(z_i - E(Z))^{2}P(z_i)=(6 - 7.6)^{2}\times0.2+(7 - 7.6)^{2}\times0.4+(8 - 7.6)^{2}\times0+(9 - 7.6)^{2}\times0.4$ $=(-1.6)^{2}\times0.2+(-0.6)^{2}\times0.4+(0.4)^{2}\times0+(1.4)^{2}\times0.4$ $=2.56\times0.2 + 0.36\times0.4+0+1.96\times0.4$ $=0.512+0.144 + 0+0.784=1.44$

Step3: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{Var(Z)}=\sqrt{1.44}=1.20$

Answer:

$1.20$