A quality control manager at a chemical company wants to measure the average amount of cleaning fluid the company is placing in their standa

Question

A quality control manager at a chemical company wants to measure the average amount of cleaning fluid the company is placing in their standard size bottle. From previous studies, they believe their population standard deviation is 0.4 ounces. If the manager would like to be 99% confident that their estimate of the mean is within 0.05 ounces of the true mean, how large of a sample is needed?

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Rose 3 days 2022-01-11T15:24:01+00:00 1 Answer 0 views 0

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    2022-01-11T15:25:19+00:00

    Answer:

    We need a sample size of at least 425.

    Step-by-step explanation:

    We have that to find our \alpha level, that is the subtraction of 1 by the confidence interval divided by 2. So:

    \alpha = \frac{1-0.99}{2} = 0.01

    Now, we have to find z in the Ztable as such z has a pvalue of 1-\alpha.

    So it is z with a pvalue of 1-0.01 = 0.99, so z = 2.575

    Now, find the margin of error M as such

    M = z*\frac{\sigma}{\sqrt{n}}

    In which \sigma is the standard deviation of the population and n is the size of the sample.

    If the manager would like to be 99% confident that their estimate of the mean is within 0.05 ounces of the true mean, how large of a sample is needed?

    We need a sample size of at least n.

    n is found when M = 0.05, \sigma = 0.4. So

    M = z*\frac{\sigma}{\sqrt{n}}

    0.05 = 2.575*\frac{0.4}{\sqrt{n}}

    0.05\sqrt{n} = 2.575*0.4

    \sqrt{n} = \frac{2.575*0.4}{0.05}

    (\sqrt{n})^{2} = (\frac{2.575*0.4}{0.05})^{2}

    n = 424.36

    Rounding up

    We need a sample size of at least 425.

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