If the mean number of years between asbestos exposure and death is found to be 25 years with a standard deviation of 7 years in a particular

Question

If the mean number of years between asbestos exposure and death is found to be 25 years with a standard deviation of 7 years in a particular county, what is the probability that the mean number of years between asbestos exposure and death will be between 22 and 29 years in a sample of 50 people taken from that community

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Kim Chi 5 months 2021-08-25T00:11:14+00:00 1 Answers 71 views 0

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    2021-08-25T00:12:14+00:00

    Answer:

    0.9988 = 99.88% probability that the mean number of years between asbestos exposure and death will be between 22 and 29 years.

    Step-by-step explanation:

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

    Normal Probability Distribution:

    Problems of normal distributions can be solved using the z-score formula.

    In a set with mean \mu and standard deviation \sigma, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

    Central Limit Theorem

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

    For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

    The mean number of years between asbestos exposure and death is found to be 25 years with a standard deviation of 7 years in a particular county

    This means that \mu = 25, \sigma = 7

    Sample of 50:

    This means that n = 50, s = \frac{7}{\sqrt{50}}

    What is the probability that the mean number of years between asbestos exposure and death will be between 22 and 29 years?

    This is the p-value of Z when X = 29 subtracted by the p-value of Z when X = 22.

    X = 29

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

    By the Central Limit Theorem

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

    Z = \frac{29 - 25}{\frac{7}{\sqrt{50}}}

    Z = 4.04

    Z = 4.04 has a p-value of 1.

    X = 22

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

    Z = \frac{22 - 25}{\frac{7}{\sqrt{50}}}

    Z = -3.03

    Z = -3.03 has a p-value of 0.0012.

    1 – 0.0012 = 0.9988

    0.9988 = 99.88% probability that the mean number of years between asbestos exposure and death will be between 22 and 29 years.

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