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:

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

Step-by-step 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:

[tex]np=40\times 0.55=22>10\\\\n(1-p)=40\times (1-0.55)18>10[/tex]

Thus, a Normal approximation to binomial can be applied.

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

[tex]\mu=np=40\times 0.55=22\\\\\sigma=\sqrt{np(1-p)}=\sqrt{40\times 0.55\times (1-0.55)}=3.15[/tex]

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

[tex]P(X\geq 24)=P(\frac{X-\mu}{\sigma}>\frac{24-22}{3.15})\\\\=P(Z>0.63)\\\\=1-P(Z<0.63)\\\\=1-0.7357\\\\=0.2643[/tex]

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

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