Question

the movie critic liked to count the number of actors in each movie she saw.

number of actors	number of movies
54	2
56	1
68	1
84	1
91	2

x is the number of actors that a randomly chosen movie had. what is the standard deviation of x?
round your answer to the nearest hundredth.

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:

$17.93$