Question

a group of 17 students cholesterol levels are tested. suppose the probability of a student having a healthy cholesterol level is 0.30.
(a) a binomial experiment has a fixed number of trials, n. for this experiment, what is the value of n?
17
(b) what is the probability of a success?
0.30
(c) what is the probability that exactly six out of 17 students have healthy cholesterol levels? (round your answer to three decimal places.)
0.179

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Substitute the values into the formula

Here, \(n = 17\), \(k = 6\), \(p=0.3\), and \(1-p = 0.7\)

First, calculate \(C(17,6)=\frac{17!}{6!(17 - 6)!}=\frac{17!}{6!×11!}=\frac{17\times16\times15\times14\times13\times12}{6\times5\times4\times3\times2\times1}=12376\)

Then, \(p^{k}=(0.3)^{6}\) and \((1 - p)^{n - k}=(0.7)^{11}\)

\(P(X = 6)=12376\times(0.3)^{6}\times(0.7)^{11}\)

\((0.3)^{6}=0.000729\)

\((0.7)^{11}\approx0.0197732674\)

\(P(X = 6)=12376\times0.000729\times0.0197732674\)

\(P(X = 6)\approx12376\times0.0000144147\)

\(P(X = 6)\approx0.178\)

Answer:

\(0.178\)