Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling mach

Question

Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.4 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains between 12.3 and 12.36 ounces.

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Autumn 2 hours 2021-10-13T01:18:08+00:00 1 Answer 0

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    2021-10-13T01:19:12+00:00

    Answer:

    15.25% probability that the bottle contains between 12.3 and 12.36 ounces.

    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 = 12.4, \sigma = 0.04

    Find the probability that the bottle contains between 12.3 and 12.36 ounces.

    This is the pvalue of Z when X = 12.36 subtracted by the pvalue of Z when X = 12.3

    X = 12.36

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

    Z = \frac{12.36 - 12.4}{0.04}

    Z = -1

    Z = -1 has a pvalue of 0.1587

    X = 12.3

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

    Z = \frac{12.3 - 12.4}{0.04}

    Z = -2.5

    Z = -2.5 has a pvalue of 0.0062

    0.1587 – 0.0062 = 0.1525

    15.25% probability that the bottle contains between 12.3 and 12.36 ounces.

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