L asked in Science & MathematicsMathematics · 4 weeks ago

If X is a normal variable with mean 10 and SD 2, then what's the area under the normal curve of X bounded by the interval 7 and 13?

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  • rotchm
    Lv 7
    4 weeks ago

    Many ways to do this. Here are two:

    A) Transform your values to z space via z = (x - avg)/sd.

    So the 7 becomes ? 

    And the 13 becomes ?

    Now look up in your z table for the area between these two. 

    This usually requires to look up the individual areas & subtracting them.

    What do you get?

    B) Note that 7 & 13 are each 1.5 SD from the avg. You can apply 

    an extension of the 68–95–99.7_rule. Wiki it. You will see that 

    μ ± 1.5σ is 0.8664.

    Done!

  • 4 weeks ago

    You can determine this with a z-score table. 

    The area under a normal curve is 1.  So the area of a portion of that will be less than one. 

    First we need the z-scores of your two endpoints:

    n = m + sz

    Where n is the data point in question (7 and 13)

    m is the mean (10)

    s is the standard deviation (2)

    z is the z-score (unknown)

    7 = 10 + 2z and 13 = 10 + 2z

    -3 = 2z and 3 = 2z

    -1.5z and 1.5 = z

    Using these points in a z-score table gives you the probability of a random data point being less than that point (aka, the area under the curve from -inf to that point).

    So if you find the area under z = 1.5 and subtract it from the area under z = -1.5 that overlaps, the result is the area between the points.

    P(z = 1.5) - P(z = -1.5)

    0.9332 - 0.0668

    0.8664

    0.8664 unit² is the answer to your question

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