Respuesta :
Using the binomial distribution, it is found that there is a 0.3154 = 31.54% probability that among the students in the sample at least 6 are female.
What is the binomial distribution formula?
The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
In this problem:
- The student body of a large university consists of 60% female students, hence p = 0.6.
- A random sample of 8 students is selected, hence n = 8.
The probability that among the students in the sample at least 6 are female is given by:
[tex]P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8)[/tex]
In which:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 6) = C_{8,6}.(0.6)^{6}.(0.4)^{2} = 0.2090[/tex]
[tex]P(X = 7) = C_{8,7}.(0.6)^{7}.(0.4)^{1} = 0.0896[/tex]
[tex]P(X = 8) = C_{8,8}.(0.6)^{8}.(0.4)^{0} = 0.0168[/tex]
Then:
[tex]P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) = 0.2090 + 0.0896 + 0.0168 = 0.3154[/tex]
0.3154 = 31.54% probability that among the students in the sample at least 6 are female.
More can be learned about the binomial distribution at https://brainly.com/question/24863377
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