91Ó°ÊÓ

75% of peoples from the metropolitan area live in the city.

25% of the people live in the suburbs.

The total number of people X from the suburbs attending the concert can be regarded as the sum of 1200 independent random variables, each of which has a Bernoulli distribution with parameter\(p = \frac{1}{4}\)and\(q = 1 - p\)

Therefore, the distribution of X will be approximately normal distribution with mean,

\(\begin{array}{c}np = 1200\left( {\frac{1}{4}} \right)\\ = 300\end{array}\)

and the variance,

\(\begin{array}{c}npq = 1200\left( {\frac{1}{4}} \right)\left( {\frac{3}{4}} \right)\\ = 225\end{array}\)

Let\(Z = \frac{{\left( {X - 300} \right)}}{{15}}\)

Then the distribution of Z will be approximately a standard normal distribution.

Hence,

\(\begin{array}{c}{\rm P}\left( {H < 270} \right) = {\rm P}\left( {X \le 269.5} \right)\\ = {\rm P}\left( {Z \le \frac{{269.5 - 300}}{{15}}} \right)\end{array}\)

\({\rm P}\left( {H < 270} \right) \approx 1 - \phi \left( {2.033} \right)\)

\({\rm P}\left( {H < 270} \right) \approx 0.0210\)

Therefore,Probability that the number of people from the suburbs attending the concert will be fewer than 270 is\(0.0210\).