91影视

The square root of the variance is the standard deviation of a random variable, sample, statistical population, data collection, or probability distribution. It is less resilient in practice than the average absolute deviation, but it is algebraically easier.

The time taken has a normal distribution with a standard deviation of \({\rm{\sigma = 2}}\)minutes and a \({\rm{\mu = 10}}\)minute mean. \({{\rm{\mu }}_{{\rm{\bar X}}}}{\rm{ = 10}}\)is the expected value of random variable \({\rm{X}}\) (sample average), and the standard deviation is

\(\begin{array}{*{20}{c}}{{{\rm{\sigma }}_{{\rm{\bar X}}}}{\rm{ = }}\frac{{\rm{1}}}{{\sqrt {\rm{n}} }}{\rm{ \times \sigma = }}\frac{{\rm{1}}}{{\sqrt {\rm{5}} }}{\rm{ \times 2 = 0}}{\rm{.894,}}\;\;\;{\rm{\;when\;n = 5}}}\\{{{\rm{\sigma }}_{{\rm{\bar X}}}}{\rm{ = }}\frac{{\rm{1}}}{{\sqrt {\rm{n}} }}{\rm{ \times \sigma = }}\frac{{\rm{1}}}{{\sqrt {\rm{6}} }}{\rm{ \times 2 = 0}}{\rm{.816,}}\;\;\;{\rm{\;when\;n = 6}}}\end{array}\)

For the first day (\({\rm{n = 5}}\)individuals), the requested probability is

\(\begin{aligned}{{{\rm{P}}_{\rm{1}}}{\rm{(\bar X拢 11) &= P\left( {\frac{{{\rm{\bar X - }}{{\rm{\mu }}_{{\rm{\bar X}}}}}}{{{{\rm{\sigma }}_{{\rm{\bar X}}}}}}{\rm{拢 }}\frac{{{\rm{11 - 10}}}}{{{\rm{0}}{\rm{.894}}}}} \right){\rm{ = P(Z拢 1}}{\rm{.12)}}}\\&= {\rm{0}}{\rm{.8686,}}}\end{aligned}\)

(1): from the appendix's normal probability table. Software can also be used to calculate the likelihood.

For the second day (\({\rm{n = 6}}\)individuals), the requested probability is

\(\begin{aligned}{{{\rm{P}}_{\rm{2}}}{\rm{(\bar X拢 11) &= P\left( {\frac{{{\rm{\bar X - }}{{\rm{\mu }}_{{\rm{\bar X}}}}}}{{{{\rm{\sigma }}_{{\rm{\bar X}}}}}}{\rm{拢 }}\frac{{{\rm{11 - 10}}}}{{{\rm{0}}{\rm{.816}}}}} \right){\rm{ = P(Z拢 1}}{\rm{.22)}}}\\ &= {\rm{0}}{\rm{.8888,}}}\end{aligned}\)

(1): from the appendix's normal probability table. Software can also be used to calculate the likelihood.

\(\begin{aligned} P(\bar X拢 11) &= {{\rm{P}}_{\rm{1}}}{\rm{(\bar X拢 11) \times }}{{\rm{P}}_{\rm{2}}}{\rm{(\bar X拢 11)}}}\\ &= 0 {\rm{.8686 \times 0}}{\rm{.8888}}}\\&= 0 {\rm{.772}}}\end{aligned}\)

Because the two days' outcomes are unrelated, the request probability (for both days) is the product of the computed probabilities.