Question

A statistics class has a population of 140 students of which 55% are female. A random sample of 40 students was selected. What is the probability that 24 or more from this sample will be female?

Answer 1

The probability that 24 or more from this sample will be female is 0.2643.

Explanation:

Define X as the number of female students.

The random variable X follows a Binomial distribution with parameters n = 40 and p = 0.55.

But the sample selected is too large and the probability of success is close to 0.50.

So a Normal approximation to binomial can be applied to approximate the distribution of X if the following conditions are satisfied:

1. np ≥ 10

2. n(1 - p) ≥ 10

Check the conditions as follows:

`np=40* 0.55=22>10\\\\n(1-p)=40* (1-0.55)18>10`

Thus, a Normal approximation to binomial can be applied.

So, X follows normal distribution with mean and standard deviation:

`\mu=np=40* 0.55=22\\\\\sigma=√(np(1-p))=√(40* 0.55* (1-0.55))=3.15`

Compute the probability that 24 or more from this sample will be female as follows:

`P(X\geq 24)=P((X-\mu)/(\sigma)>(24-22)/(3.15))\\\\=P(Z>0.63)\\\\=1-P(Z<0.63)\\\\=1-0.7357\\\\=0.2643`

Thus, the probability that 24 or more from this sample will be female is 0.2643.