Question

Suppose that 1000 people are randomly selected and tested for a particular disease that is present in 1.2% of the population. Assume that the test will perfectly identify the disease presence/abscence in each tested individual.

For the following, for each numerical response, round your answer to at least 4 decimals(when necessary).

-Use a binomial distribution to find the probability that at least 20 people test positive for the disease.

-Approximate the previous probabilty by using the normal distribution.

-Use a Poisson distribution to find the probability that at least 20 people test positive for the disease.

-Approximate the previous probability by using the normal distribution.

Answer 1

i) The probability that any individual has the disease = 1.2% = 0.012.

The pdf of a binomial distribution is given by

The probability that at least 20 people test positive for the disease is given by

The probability that at least 20 people test positive for the disease is = 0.021

ii) Using a normal approximation to the binomial distribution, we have

The average number of individuals getting the disease = 1.2% of 1000 = 12

The variance of the number of individuals getting the disease

= np(1-p)

= 1000 * 0.012 * 0.988

= 11.856

The probability that at least 20 individuals will test positive for the disease is given by

Using the normal approximation, the probability that at least 20 individuals will test positive for the disease is= 0.01

iii) Using a Poisson's distribution

The average number of individuals getting the disease = 1.2% of 1000 = 12

The pdf of a Poisson's distribution is given by

The probability that at least 20 individuals will get the disease

= 1 - The probability that at most 19 people will get the disease

Hence, using a Poisson's distribution, we have

The probability that at least 20 individuals will get the disease = 0.021

iv) Using normal approximation to Poisson's distribution

The average number of individuals getting the disease = 1.2% of 1000 = 12

The variance of the number of individuals getting the disease= 12

The probability that at least 20 individuals will test positive for the disease is given by

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