Question

Suppose a simple random sample of size n=1000 is obtained from a population whose size is N=2,000,000 and whose population proportion with a specified characteristic is p = 0.22

a. What is the probability of obtaining x=250 or more individuals with the​ characteristic?

Answer 1

Given:

Let X denote the random variable with the specific characteristic.

Based on the given data,

Since, computing the exact probability for this large a sample is a tedious task, we may use a normal approximation.

For n = 1000 (large), using Central limit theorem which states that, for large n,

But the standard normal variate is continuous, while the binomial variable X is discrete; Before applying the central limit theorem, we need to apply the continuity correction, by expressing X as an interval, .i.e. X = 250 can be expressed as (249.5,250.5)

Since we are interested in greater than probability , we may use the upper limit of the interval for further computations.

Looking for the probability using standard normal table,

= 1 - 0.9901

= 0.0099

Hence, the probability of obtaining x=250 or more individuals with the​ characteristic is approximately 0.01 (negligible).

To find the exact probability,

Using excel,

Using the formula,

we get

And, the formula

gives P(X = 250) = 0.00229

Hence,

= 1 - [ ]

= 1 - [ 0.98925 - 0.00229 ]

= 0.01304

Hence, the probability of obtaining x=250 or more individuals with the​ characteristic is 0.01304.