EX 4.20

• Suppose that
  • $25\%$ of all students at a large public university receive financial aid
  • $X$ be the number of students in a random sample of size 50 who receive financial aid,
    • so that $p=.25$ .
• Since
  • $np=50\left(.25\right)=12.5\ge 10$
  • $nq=37.5\ge 10$
• the approximation can safely be applied.
  • $\mu =12.5$
  • $\sigma =3.06$ .
• The probability that at most 10 students receive aid is

$$\begin{array}{rl}& P\left(X\le 10\right)\\ & =B\left(10;50,0.25\right)\\ & \approx \Phi \left(\frac{10+0.5-12.5}{3.06}\right)\\ & =\Phi \left(-0.65\right)\\ & =0.2578\end{array}$$

• Similarly, the probability that between 5 and 15 (inclusive) of the selected students receive aid is

$$\begin{array}{rl}& P\left(5\le X\le 15\right)\\ & =B\left(15;50,0.25\right)-B\left(4;50,0.25\right)\\ & \approx \Phi \left(\frac{15.5-12.5}{3.06}\right)-\Phi \left(\frac{4.5-12.5}{3.06}\right)\\ & =0.8320\end{array}$$

• The exact probabilities are .2622 and .8348 , respectively,
  • so the approximations are quite good.
• In the last calculation, $P\left(5\le X\le 15\right)$ is being approximated by the area under the normal curve between 4.5 and 15.5 - the continuity correction is used for both the upper and lower limits.