A manufacturer knows that their items have a normally distributed lifespan, with a mean of 5.5 years, and standard deviation of 1.2 years. I

Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 5.5 years, and standard deviation of 1.2 years. If 24 items are picked at random, 8% of the time their mean life will be less than how many years?

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Jasmine 3 weeks 2021-09-21T13:06:04+00:00 1 Answer 0

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    2021-09-21T13:07:29+00:00

    Answer:

    If 24 items are picked at random, 8% of the time their mean life will be less than 5.156 years.

    Step-by-step explanation:

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

    Normal probability distribution:

    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.

    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 = 5.5, \sigma = 1.2, n = 24, s = \frac{1.2}{\sqrt{24}} = 0.2449

    If 24 items are picked at random, 8% of the time their mean life will be less than how many years?

    This is the value of X when Z has a pvalue of 0.08. So it is X when Z = -1.405.

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

    By the Central Limit Theorem

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

    -1.405 = \frac{X - 5.5}{0.2449}

    X - 5.5 = -1.405*0.2449

    X = 5.156

    If 24 items are picked at random, 8% of the time their mean life will be less than 5.156 years.

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