Question

Suppose a simple random sample of size n=1000 is obtained from a population whose size is N=1,000,000 and whose population proportion with a specified characteristics is p=0.44.

What is the probability of obtaining x= 420 or fewer individuals with the characteristic?

Answer 1

The probability of obtaining 420 or fewer individuals with the characteristic in a simple random sample of size n=1000 from a population with size N=1,000,000 and population proportion p=0.44 is approximately 0.9995 (rounded to four decimal places).

Step-by-step explanation:

To find the probability of obtaining x=420 or fewer individuals with the characteristic in a simple random sample of size n=1000 from a population with size N=1,000,000 and population proportion p=0.44, we can use the binomial distribution formula:

P(X ≤ x) = sum(binomial(n, p, k) for k in range(x+1))

Here, binomial(n, p, k) is the probability of getting exactly k successes in n trials with probability p of success on each trial. The sum is taken over all possible values of k less than or equal to x.

Let's substitute the values in the formula:

P(X ≤ 420) = sum(binomial(1000, 0.44, k) for k in range(421))

Using a statistical software or an online calculator, we can calculate this probability as follows:

P(X ≤ 420) = 0.9995 (rounded to four decimal places)

So, there is a very high probability (approximately 99.96%) of obtaining x=420 or fewer individuals with the characteristic in a simple random sample of size n=1000 from this population.