A biologist estimates that $60 \%$ 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 186 or fewer deer have this tick?
A) 0.8135
B) 0.7782
C) 0.7344
D) 0.7290

Question

Quizplus · Accepted Answer

This problem involves a binomial distribution with parameters n=300 and p=0.6. We want to find the probability that 186 or fewer deer have the tick, which can be expressed as:

P(X ≤ 186) = Σ P(X = k) for k = 0 to 186

Using a binomial distribution table or calculator, we can find the probabilities for each value of k and then add them up. However, this can be time-consuming and tedious.

Alternatively, we can use a normal approximation to the binomial distribution, which is valid for large n and for probabilities not too close to 0 or 1. We can approximate the binomial distribution with a normal distribution with mean μ=np=180 and variance σ²=np(1-p)=72. Using the continuity correction, we have:

P(X ≤ 186) ≈ P(Z ≤ (186.5-180)/sqrt(72)) = P(Z ≤ 0.906)

where Z is a standard normal random variable. This probability can be found using a standard normal table or calculator, which gives:

P(Z ≤ 0.906) ≈ 0.1782

To find the probability that 186 or fewer deer have the tick, we need to subtract this probability from 1:

P(X ≤ 186) ≈ 1 - 0.1782 = 0.8218

This probability is closest to B) 0.7782, which is the correct answer.

A Biologist Estimates That 60%60 \%60% Of Deer in the Region Carry a Certain Type of Region

A Biologist Estimates That $60 \%$ Of Deer in the Region Carry a Certain Type of Region