Answer to Question #190638 in Statistics and Probability for Mayah

Question

a biologist estimates that 50 of deer in the region carry a certain type of tick. For a sample of 300 deer selected at random, what is the chance that 155 or fewer deer have this tick?

Answer 1

Let X denote number of deers in the region carry a certain type of tick

Here X follows binomial distribution with parameters $n=300 \text{ and } p=0.16$

From the normal approximation to the binomial variable-

$z=\dfrac{x-np}{\sqrt{np(1-p)}}$ is the standard normal variable.

Probability that 155 or fewer deer have the tick-

$P(X\le 155)=P(X\le 155+0.5)=P(X<155.5)$

 $=P(z\le \dfrac{155.5-(300)(0.16)}{\sqrt{300(0.16(1-0.16)}})\\[9pt]=P(z\le \dfrac{107.5}{6.349})\\[9pt]=P(z\le 16.93)$

   $=P(Z\le 16.93)=0.7782$