A certain college would like to have 1050 freshmen. This college cannot accommodate more than 1060. Assume that each applicant accepts with

Question

A certain college would like to have 1050 freshmen. This college cannot accommodate more than 1060. Assume that each applicant accepts with probability .6 and that the acceptances can be modeled by a Binomial distribution. If the college takes 1700 students, what is the probability that the college will have too many acceptances (that is, more than 1060)

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Amaya 4 weeks 2021-09-19T03:03:54+00:00 1 Answer 0

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    2021-09-19T03:05:26+00:00

    Answer:

    2.39% probability that the college will have too many acceptances

    Step-by-step explanation:

    We use the binomial approximation to the normal to solve this question.

    For each item selected, there are only two possible outcomes. Either it is defective, or it is not. This means that we use concepts of the binomial probability distribution to solve this problem.

    However, we are working with samples that are considerably big. So i am going to aproximate this binomial distribution to the normal.

    Binomial probability distribution

    Probability of exactly x sucesses on n repeated trials, with p probability.

    Can be approximated to a normal distribution, using the expected value and the standard deviation.

    The expected value of the binomial distribution is:

    E(X) = np

    The standard deviation of the binomial distribution is:

    \sqrt{V(X)} = \sqrt{np(1-p)}

    Normal probability distribution

    Problems of normally distributed samples can be solved using the z-score formula.

    In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

    Z = \frac{X - \mu}{\sigma}

    The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

    When we are approximating a binomial distribution to a normal one, we have that \mu = E(X), \sigma = \sqrt{V(X)}.

    In this problem, we have that:

    n = 1700, p = 0.6

    So

    \mu = E(X) = np = 1700*0.6 = 1020

    \sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1700*0.6*0.4} = 20.2

    If the college takes 1700 students, what is the probability that the college will have too many acceptances (that is, more than 1060)

    This is 1 subtracted by the pvalue of Z when X = 1060. So

    Z = \frac{X - \mu}{\sigma}

    Z = \frac{1060 - 1020}{20.2}

    Z = 1.98

    Z = 1.98 has a pvalue of 0.9761

    1 – 0.9761 = 0.0239

    2.39% probability that the college will have too many acceptances

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