You take a simple random sample of 100 adults from a town in the Western United States to determine the proportion of those in town who inve

Question

You take a simple random sample of 100 adults from a town in the Western United States to determine the proportion of those in town who invest in the stock market. Assume the unknown population proportion or percentage of people in town who invest in the stock market is p = 0.30 (or 30%).
Part 1: What is the mean of the distribution of the sample proportions?

A. 30
B. 70
C. 0.70
D. 0.30

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Caroline 4 days 2021-10-09T18:56:59+00:00 1 Answer 0

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    2021-10-09T18:58:55+00:00

    Answer:

    A. 30

    Step-by-step explanation:

    1) Previous concepts

    The binomial distribution is a “DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other”.

    The Sampling distribution of proportion \hat p “is a collection of repeated samples of given size n taken from the same population. The mean of all these samples proportions is the population proportion”.

    2) Solution to the problem

    Let X the random variable of interest, on this case we assume that the distribution for this random variable is given by that:

    X \sim Binom(n=100, p=0.3)

    The probability mass function for the Binomial distribution is given as:

    P(X)=(nCx)(p)^x (1-p)^{n-x}

    Where (nCx) means combinatory and it’s given by this formula:

    nCx=\frac{n!}{(n-x)! x!}

    The expected vaue for this distribution or mean is given by:

    E(x)]= \mu =np=100*0.3=30

    And the variance is given by:

    Var(X) =np(1-p)=100*0.3*(1-0.3)=21

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