Question

The Office for admission of Hero Academy found out the IQ of incoming 1,200 freshmen normally distributed with a mean of 108 and a standard deviation of 10. They make a table to interpret their IQ.

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} IQ\ Score & Interpretation \\ \hline >130 & Very \ Gifted \\ 121-130 & Gifted \\ 111-120 & Above \ Average\ Intelligance \\ 90-110 & Average\ Intelligance \\ 80-89 & Below\ Average\ Intelligance \\ 70-79 & Cognitively \ Impaired \\ \hdashline \end{array}$$

1. How many students have an average IQ or lower

$$P(X\leq110)=P(Z\leq\dfrac{110-108}{10})=$$

$$=P(Z\leq0.2)\approx0.57925971$$

$0.03593032\cdot1200\approx695$

695 students have an average IQ or lower.

2. If all gifted and very gifted students will be given scholarships, how much budget is needed if each scholar is expected to receive Php 20,000 per year?

$$P(X\geq121)=1-P(X<121)=$$

$$=1-P(Z<\dfrac{121-108}{10})=1-P(Z<1.3)\approx$$

$$\approx1-0.90319952\approx0.09680048$$

$0.09680048\cdot1200\approx116$

$116\cdot20000=2,320,000$

Php 2,320,000 per year is needed.