In a study, 42% of adults questioned reported that their health was excellent. A researcher wishes to study the health of people living clos

Question

In a study, 42% of adults questioned reported that their health was excellent. A researcher wishes to study the health of people living close to a nuclear power plant. Among 11 adults randomly selected from this area, only 3 reported that their health was excellent. Find the probability that when 11 adults are randomly selected, 3 or fewer are in excellent health.

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Alice 3 weeks 2021-11-20T11:44:49+00:00 1 Answer 0 views 0

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    2021-11-20T11:46:09+00:00

    Answer:

    The probability that when 11 adults are randomly selected, 3 or fewer are in excellent health is 0.6483.

    Step-by-step explanation:

    The random variable X can be defined as the number of adults questioned reported that their health was excellent.

    A random sample of n = 11 adults are selected randomly selected from an area a nuclear power plant.

    Of these 11 adults X = 3 reported that their health was excellent.

    The proportion of adults who reported that their health was excellent is:

    p=\frac{X}{n}=\frac{3}{11}

    An adult’s health condition is not affected by others, i.e. they are independent.

    The random variable X follows a Binomial distribution with parameters n = 11 and p = \frac{3}{11}.

    The probability mass function of X is:

    P(X=x)={11\choose x}(\frac{3}{11})^{x}(1-\frac{3}{11})^{11-x};\ x=0,1,2,3...

    Compute the probability that when 11 adults are randomly selected, 3 or fewer are in excellent health as follows:

    P (X ≤ 3) = P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)

                  =\sum\limits^{3}_{x=0}{{11\choose x}(\frac{3}{11})^{x}(1-\frac{3}{11})^{11-x}}\\=0.02998+0.12385+0.23254+0.26197\\=0.64834\\\approx 0.6483

    Thus, the probability that when 11 adults are randomly selected, 3 or fewer are in excellent health is 0.6483.

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