The area under each curve is one but the scaling of the X axis is different. Note, however, that the areas to the left of the dashed line are the same. The BMI distribution ranges from 11 to 47, while the standardized normal distribution, Z, ranges from -3 to 3. We want to compute P(X < 30). To do this we can determine the Z value that corresponds to X = 30 and then use the standard normal distribution table above to find the probability or area under the curve. The following formula converts an X value into a Z score, also called a standardized score:
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where ÎĽ is the mean and Ď is the standard deviation of the variable X.
In order to compute P(X < 30) we convert the X=30 to its corresponding Z score (this is called standardizing):
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Thus, P(X < 30) = P(Z < 0.17). We can then look up the corresponding probability for this Z score from the standard normal distribution table, which shows that P(X < 30) = P(Z < 0.17) = 0.5675. Thus, the probability that a male aged 60 has BMI less than 30 is 56.75%.
