Suppose that in a large population the proportion of people that have a certain disease

is 0.01. Use the Poisson approximation to find the probability that at least four people will
have the disease in a group of 400 randomly selected people.

lol are you taking stat 2507 were all in the same boat

SAME CLASS FML

X~Bin(400,0.01)

np= 4
P(x>=4)= 1-P(Y<=3)
1-[P(X=3)+P(X=2)+P(X=1)+P(X=0)]
1- ((e^-4)*(4^3))/3! +((e^-4)*(4^2))/2! + ((e^-4)*(4^1))/1! +((e^-4)*(4^0))/0!=
1- 0.4330= 0.567

To find the probability that at least four people will have the disease in a group of 400 randomly selected people using the Poisson approximation, we can follow these steps:

1. Understand the problem: We want to find the probability of an event (four or more people having the disease) occurring in a large population with a known proportion of people with the disease (0.01) and a sample size of 400.

2. Apply the Poisson approximation: The Poisson distribution can be used to approximate the probability of rare events occurring, such as the occurrence of disease cases in this scenario.

3. Calculate the mean: The mean of a Poisson distribution is equal to the product of the population size (n) and the proportion of individuals with the disease (p). In this case, n = 400 and p = 0.01, so the mean (λ) is given by λ = n * p.

λ = 400 * 0.01
λ = 4

4. Apply the Poisson formula: The probability mass function of the Poisson distribution is given by the formula P(X = k) = (e^(-λ) * λ^k) / k!, where X is the random variable representing the number of disease cases, k is the number of disease cases we want to calculate the probability for, and λ is the mean.

P(X ≥ 4) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3))

P(X = k) can be calculated using the Poisson formula.

5. Calculate the probability: Substituting the values into the formula, we can find the individual probabilities and calculate their sum to find the probability of at least four people having the disease.

P(X = k) = (e^(-λ) * λ^k) / k!
P(X ≥ 4) = 1 - (e^(-4) * 4^0 / 0! + e^(-4) * 4^1 / 1! + e^(-4) * 4^2 / 2! + e^(-4) * 4^3 / 3!)

Using a calculator or statistical software, evaluate this expression to find the probability.

Note: The Poisson approximation can be used when the population size is large, and the proportion of individuals with the characteristic of interest is small (rare event). If those conditions are not met, a different approach may be required.