A large retirement community has a current population more than 5,000 residents. The distribution of ages of all residents is left skewed wi

Question

A large retirement community has a current population more than 5,000 residents. The distribution of ages of all residents is left skewed with a mean of 65.5 years and a standard deviation of 12.5 year. Suppose you are to conduct a survey and take a random sample of 100 residents of the community. Which of the following corretly describe how to find the probability that you obtain a sample mean age that is younger than 64 years? a) Find the area to the left of z = -0.12 under a standard normal curve. b) Find the area to the right of z = -1.2 under a standard normal curve. c) Find the area to the left of z = -1.2 under a standard normal curve. d) Find the area to the right of z = -0.12 under a standard normal curve. e) None of above

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Adalynn 2 days 2021-10-12T22:10:53+00:00 1 Answer 0

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    2021-10-12T22:12:04+00:00

    Answer:

    c) Find the area to the left of z = -1.2 under a standard normal curve.

    Step-by-step explanation:

    To solve this question, we need to understand the normal probability distribution and the central limit theorem.

    Normal probabiliy distribution

    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, or the area to the left of Z in the normal curve. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X, which is the area to the right of Z in the normal curve.

    Central limit theorem:

    The Central Limit Theorem estabilishes that, for a random variable X, with mean \mu and standard deviation \sigma, the sample means with size n of at least 30 can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}

    In this problem, we have that:

    \mu = 65.6, \sigma = 12.5, n = 100, s = \frac{12.5}{\sqrt{100}} = 1.25

    Which of the following corretly describe how to find the probability that you obtain a sample mean age that is younger than 64 years?

    Area to the left of z when X = 64 under the standard normal curve. We have to find Z.

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

    Applying the Central Limit Theorem

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

    Z = \frac{64 - 65.6}{1.25}

    Z = -1.2

    So the correct answer is:

    c) Find the area to the left of z = -1.2 under a standard normal curve.

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