3.30 Survey response rate. Pew Research reported in 2012 that the typical response rate to their surveys is only 9%. If for a particular sur

Question

3.30 Survey response rate. Pew Research reported in 2012 that the typical response rate to their surveys is only 9%. If for a particular survey 15,000 households are contacted, what is the probability that at least 1,500 will agree to respond

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Savannah 1 month 2021-10-17T09:29:31+00:00 1 Answer 0 views 0

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    2021-10-17T09:31:21+00:00

    Answer:

    0% probability that at least 1,500 will agree to respond

    Step-by-step explanation:

    I am going to use the binomial approximation to the normal to solve this question.

    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 = 15000, p = 0.09

    So

    \mu = E(X) = np = 15000*0.09 = 1350

    \sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{15000*0.09*0.91} = 35.05

    What is the probability that at least 1,500 will agree to respond

    This is 1 subtracted by the pvalue of Z when X = 1500-1 = 1499. So

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

    Z = \frac{1499 - 1350}{35.05}

    Z = 4.25

    Z = 4.25 has a pvalue of 1.

    1 – 1 = 0

    0% probability that at least 1,500 will agree to respond

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45:7+7-4:2-5:5*4+35:2 =? ( )