On a standard IQ test, the standard deviation is 15. How many random IQ scores must be obtained if we want to find the true population mean

Question

On a standard IQ test, the standard deviation is 15. How many random IQ scores must be obtained if we want to find the true population mean (with an allowable error of 0.5) and we want 97 percent confidence in the results

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Serenity 2 weeks 2021-09-15T09:30:11+00:00 1 Answer 0

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    2021-09-15T09:32:01+00:00

    Answer:

    We need at least 4238 scores.

    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{1-0.97}{2} = 0.015

    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 1-0.015 = 0.985, so z = 2.17

    Now, find the margin of error M

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

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

    How many random IQ scores must be obtained if we want to find the true population mean (with an allowable error of 0.5) and we want 97 percent confidence in the results

    We need at least n scores, in which N is found when M = 0.5. So

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

    0.5 = 2.17*\frac{15}{\sqrt{n}}

    0.5\sqrt{n} = 2.17*15

    0.5\sqrt{n} = 32.55

    \sqrt{n} = \frac{32.55}{0.5}

    \sqrt{n} = 65.1

    \sqrt{n}^{2} = (65.1)^{2}

    n = 4238

    We need at least 4238 scores.

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