the movie critic liked to count the number of actors in each movie she saw.\n|number of actors|number of…

the movie critic liked to count the number of actors in each movie she saw.\n|number of actors|number of movies|\n|----|----|\n|45|3|\n|54|2|\n|56|1|\n|68|1|\n|84|1|\n|91|2|\nx is the number of actors that a randomly chosen movie had. what is the standard deviation of x?\nround your answer to the nearest hundredth.

the movie critic liked to count the number of actors in each movie she saw.\n|number of actors|number of movies|\n|----|----|\n|45|3|\n|54|2|\n|56|1|\n|68|1|\n|84|1|\n|91|2|\nx is the number of actors that a randomly chosen movie had. what is the standard deviation of x?\nround your answer to the nearest hundredth.

Answer

Explanation:

Step1: Calculate the total number of movies

$3 + 2+1 + 1+1 + 2=10$

Step2: Calculate the expected - value (mean) $\mu$

[ \begin{align*} \mu&=\frac{45\times3 + 54\times2+56\times1 + 68\times1+84\times1 + 91\times2}{10}\ &=\frac{135+108 + 56+68+84+182}{10}\ &=\frac{633}{10}=63.3 \end{align*} ]

Step3: Calculate the variance $\sigma^{2}$

[ \begin{align*} \sigma^{2}&=\frac{3\times(45 - 63.3)^{2}+2\times(54 - 63.3)^{2}+1\times(56 - 63.3)^{2}+1\times(68 - 63.3)^{2}+1\times(84 - 63.3)^{2}+2\times(91 - 63.3)^{2}}{10}\ &=\frac{3\times(- 18.3)^{2}+2\times(-9.3)^{2}+1\times(-7.3)^{2}+1\times(4.7)^{2}+1\times(20.7)^{2}+2\times(27.7)^{2}}{10}\ &=\frac{3\times334.89+2\times86.49+1\times53.29+1\times22.09+1\times428.49+2\times767.29}{10}\ &=\frac{1004.67+172.98+53.29+22.09+428.49+1534.58}{10}\ &=\frac{3216.1}{10}=321.61 \end{align*} ]

Step4: Calculate the standard - deviation $\sigma$

$\sigma=\sqrt{321.61}\approx17.93$

Answer:

$17.93$