Question

The percentage of the Illinois population with diabetes is believed to be 9.1%.

B1) Suppose we decided to take a random sample of 20 people from the Illinois population.

1. What is the probability that none of them have diabetes? (2 pts)
2. What is the probability that at least 20% of the sample has diabetes? (2 pts)

B2) Suppose we decided to take a random sample of 200 people from the Illinois population.

1. How many people would we expect to have diabetes in the sample? (2 pts)
2. What is the probability that no less than 20 people and no more than 30 people in the sample have diabetes? (2 pts)

Answer 1

B1)

Let X is a random variable shows the number of people has diabetes out of 20. Here X has binomial distribution with parameters n=20 and p=0.091.

The  probability that none of them have diabetes is

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20% of the sample means 0.20* 20 = 4

So the probability that at least 20% of the sample (that is at least 4) has diabetes is

B2)

Here we have

n=200, p=0.091

The expected number of people is

200 * 0.091 = 18.2

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The required probability is

The excel function used: "=BINOMDIST(30,200,0.091,TRUE)-BINOMDIST(19,200,0.091,TRUE)"