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30 June, 11:40

The comprehensive strength of concrete is normally distributed with μ = 2500 psi and σ = 50 psi. Find the probability that a random sample of n = 5 specimens will have a sample mean diameter that falls in the interval from 2499 psi to 2510 psi. Express the final answer to three decimal places (e. g. 0.987).

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  1. 30 June, 11:56
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    The probability that a random sample of n = 5 specimens will have a sample values that falls in the interval from 2499 psi to 2510 psi = P (2499 < x < 2510) = 0.192

    Step-by-step explanation:

    For the population,

    μ = 2500 psi and σ = 50 psi

    But for a sample of n = 5

    μₓ = μ = 2500 psi

    σₓ = σ/√n = (50/√5)

    σₓ = 22.36 psi

    So, probability that the value for the sample falls between 2499 psi to 2510 psi

    P (2499 < x < 2510)

    We normalize/standardize these values firstly,

    The standardized score for any value is the value minus the mean then divided by the standard deviation.

    For 2499 psi

    z = (x - μ) / σ = (2499 - 2500) / 22.36 = - 0.045

    For 2510 psi

    z = (x - μ) / σ = (2510 - 2500) / 22.36 = 0.45

    To determine the probability the value for the sample falls between 2499 psi to 2510 psi

    P (2499 < x < 2510) = P (-0.045 < z < 0.45)

    We'll use data from the normal probability table for these probabilities

    P (2499 < x < 2510) = P (-0.045 < z < 0.45) = P (z < 0.45) - P (z < - 0.045) = 0.674 - 0.482 = 0.192
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