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  3. Six hundred applicants for several post office jobs were rated on a scale from 1 to 50 points. The ratings had a mean of 32.5 po

Six hundred applicants for several post office jobs were rated on a scale from 1 to 50 points. The ratings had a mean of 32.5 po

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问题 Six hundred applicants for several post office jobs were rated on a scale from 1 to 50 points. The ratings had a mean of 32.5 points and a standard deviation of 7.1 points. How many standard deviations above or below the mean is a rating of 48 points? A rating of 30 points? A rating of 20 points?
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答案48 points is 15.5 points above the mean, or approximately 2.2 standard deviations above the mean. 30 points is 2.5 points below the mean, or approximately 0.4 standard deviation below the mean. 20 points is 12.5 points below the mean, or approximately 1.8
解析 Let d be the standard deviation, so d = 7.1 points. Note that 1 standard deviation above the mean is
32.5 + d = 32.5 + 7.1 = 39.6
and 2 standard deviations above the mean is
32.5 + 2d = 32.5 + 2(7.1)= 46.7
So 48 is a little more than 2 standard deviations above the mean. Since 48 is actually 15.5 points above the mean, the number of standard deviations that 48 is above the mean is≈2.2. Thus, to answer the question, we first found the difference from the mean and then we divided by the standard deviation. The number of standard deviations that a rating of 30 is away from the mean is

where the negative sign indicates that the rating is 0.4 standard deviation below the mean.
The number of standard deviations that a rating of 20 is away from the mean is

where the negative sign indicates that the rating is 1.8 standard deviations below the mean.
To summarize:
    48 points is 15.5 points above the mean, or approximately 2.2 standard deviations above the mean.
    30 points is 2.5 points below the mean, or approximately 0.4 standard deviation below the mean.
    20 points is 12.5 points below the mean, or approximately 1.8 standard deviations below the mean.
One more instance, which may seem trivial, is important to note:
    32.5 points is 0 points from the mean, or 0 standard deviations from the mean.
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