Statistics Help?

Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.36 and a standard deviation of 1.46. Using the empirical rule, what percentage of American women have shoe sizes that are at least 11.28? Please do not round your answer.

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  • 6 months ago
    Favorite Answer

    You need to find the z-score, which can be found with:

    n = m + sz

    where:

    n = the data point you are looking for (11.28)

    m = the mean (8.36)

    s = the standard deviation (1.46)

    z = the z-score (unknown)

    Substitute what we know and solve:

    n = m + sz

    11.28 = 8.36 + 1.46z

    2.92 = 1.46z

    2 = z

    The empirical rule states that 95% of data points are within 2 standard deviations from the mean. This would include the range of:

    -2s < m < 2s

    You are asked for "at least 11.28" which means you don't want the range above, but the upper portion of what's left.

    So if we don't want the range from -2s < m < 2s, we look at the negative data points:

    x < -2s and x > 2s

    which takes up 5% of the data.

    But since we only want the values above 2s, not below -2s, we throw out that half, which leaves us with:

    2.5% of the data points being larger than 11.28 for a shoe size.

  • Mike G
    Lv 7
    6 months ago

    z(x=11.28) = (11.28-8.36)/1.46 = 2

    95/2 = 47.5

    100-50-47.5 = 2.5

    Answer 2.5%

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