) In a rural area, the infected rate of certain disease is 0.20 (i.e., 20%). A sample of 20 people is taken from the area - selections are independent. Define X as the number of people in the sample who are infected by the disease.
(1) This X is a binomial variable. What are the parameter values of this distribution? What are the mean value and standard deviation of X?
Ans.
(2) What is the probability that at least 4 of the 20 people are infected? Use Binomial table.
Ans.
(3) Compute P(X 4) using normal approximation and Poisson approximation, respectively.
Ans.
Normal approximation:
Poisson approximation:
(4) Are those in (3) good approximations? Why?
Ans.
(1) The parameter values of a binomial distribution are the number of trials (n) and the probability of success in each trial (p). In this case, the number of trials is 20 (since 20 people were sampled) and the probability of being infected is 0.20 (20%). So, the parameter values of this distribution are n = 20 and p = 0.20.
To find the mean value (μ) of X, we use the formula μ = n * p. Substituting the values, we get μ = 20 * 0.20 = 4.
To find the standard deviation (σ) of X, we use the formula σ = sqrt(n * p * (1 - p)). Substituting the values, we get σ = sqrt(20 * 0.20 * (1 - 0.20)) = sqrt(20 * 0.20 * 0.80) = sqrt(3.2) ≈ 1.79.
So, the mean value of X is 4 and the standard deviation is approximately 1.79.
(2) To find the probability that at least 4 of the 20 people are infected, we can use the binomial table. The binomial table provides the probabilities of different values of X for a given n and p.
To use the binomial table, we look for the row corresponding to n = 20 and the column corresponding to p = 0.20. Then, we sum the probabilities for X = 4, X = 5, X = 6, and so on, until X = 20.
(3) To compute P(X ≥ 4) using the normal approximation, we can use the normal distribution with mean μ = np and standard deviation σ = sqrt(np(1-p)). We approximate the binomial distribution by a normal distribution when n is large.
Using the normal distribution, we can calculate P(X ≥ 4) by finding the probability of X being greater than or equal to 4. We can use the standard normal table or a calculator with the normal distribution function to find the probability.
To compute P(X ≥ 4) using the Poisson approximation, we can use the Poisson distribution with mean μ = np. We approximate the binomial distribution by a Poisson distribution when n is large and p is small.
Using the Poisson distribution, we can calculate P(X ≥ 4) by finding the probability of X being greater than or equal to 4. We can use the Poisson distribution function in a calculator or a Poisson table to find the probability.
(4) To determine if the normal and Poisson approximations are good approximations, we need to consider the conditions under which they are valid.
For the normal approximation to be valid, n (the number of trials) should be large and both np and n(1-p) should be greater than 5. If these conditions are met, the normal approximation can be considered good.
For the Poisson approximation to be valid, n (the number of trials) should be large and p (the probability of success) should be small. If these conditions are met, the Poisson approximation can be considered good.
So, we need to check if n is large and if both np and n(1-p) are greater than 5. If these conditions are met, then both the normal and Poisson approximations can be considered good.