Question

A sleeping pill is effective for 75% of the population. If in a hospital 160 patients are given a sleeping pill, what is the approximate probability that 125 or more of them will sleep better?

Answer 1

Answer:

0.20611

Step-by-step explanation:

After taking the sleeping pills there are only two outcomes;

The pill is effective ( success), with probability p=0.75

The pill is not effective (failure), with probability 1-p=0.25

if X is the number of successes, the X is a random variable with a Bernoulli distribution (p=0.75)

Since the number of patients is large enough, we apply the law of large numbers and approximate this using a normal distribution.

$\sum{X_i$ is approximately N(np, npq)

therefore;

$\sum{X_i$ is approximately N(160*0.75, 160*0.75*0.25)

$\sum{X_i$ is approximately N(120,  30)

Required;

$P(\sum X_i \geq 125)$

since we are applying a Normal Approx on a discrete distribution, we use continuity correction as follows;

$P(\sum X_i \geq 125)=P(\sum X_i > 124.5)\\=1-P(\sum X_i \leq 124.5)\\=1-P(\frac{\sum X_i-np}{\sqrt{npq} }\leq  \frac{124.5-120}{\sqrt[2]{30} })\\ =1-P(z\leq 0.82158)\\=1-0.79389\\=0.20611$