Suppose that replacement times for washing machines are normally distributed with a mean of 8.6 years and a standard deviation of 1.6 years.

Question

Suppose that replacement times for washing machines are normally distributed with a mean of 8.6 years and a standard deviation of 1.6 years. Find the replacement time that separates the top 18% from the bottom 82%.

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Anna 2 weeks 2021-09-13T07:33:22+00:00 1 Answer 0

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    2021-09-13T07:34:58+00:00

    Answer:

    Replacement time of 10.064 years.

    Step-by-step explanation:

    Problems of normally distributed samples are 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.

    In this problem, we have that:

    \mu = 8.6, \sigma = 1.6

    Find the replacement time that separates the top 18% from the bottom 82%.

    This is the value of X when Z has a pvalue of 0.82. So it is X when Z = 0.915.

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

    0.915 = \frac{X - 8.6}{1.6}

    X - 8.6 = 0.915*1.6

    X = 10.064

    Replacement time of 10.064 years.

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