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24 March, 01:46

The mean life of a certain brand of auto batteries is 44 months with a standard deviation of 3 months. Assume that the lives of all auto batteries of this brand have a bell-shaped distribution. Using the empirical rule, find the percentage of auto batteries of this brand that have a life of 35 to 53 months. Round your answer to one decimal place.

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  1. 24 March, 01:59
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    99.7%

    Step-by-step explanation:

    Using the z score formula

    z-score is z = (x-μ) / σ

    where:

    x = raw score

    μ = population mean

    σ = population standard deviation.

    a) for x = raw score = 35

    μ = population mean = 44

    σ = population standard deviation = 3

    z = (35 - 44) / 3

    z = - 9/3

    z = - 3

    b) for x = raw score = 53

    μ = population mean = 44

    σ = population standard deviation = 3

    z = (53 - 44) / 3

    z = 9/3

    z = 3

    We would use the standard normal distribution table to find their probabilities

    P (Z<=-3) = 0.0013499

    P (Z<=+3) = 0.99865

    So P (-3<=Z<=3) = 0.99865 - 0.0013499 = 0.9973

    Converting to percentage = 0.9973 * 100 = 99.73%

    Therefore, the percentage of auto batteries of this brand that have a life of 35 to 53 months to one decimal place is 99.7%
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