Question

The serum cholesterol levels for men in one age group are normally distributed with a mean of 178.1 and a standard deviation of 40.7. (All units are in mg/100mL). Find the two levels that separate the top 9% and the bottom 9%.

Please explain!

Answer 1

For the top 9% we have to find the value x for which

P(X<x) = 0.91

Converted to the standard normal distribution this corresponds to

P(Z<(x-178.1)/40.7) = 0.91

Then: (x-178.1)/40.7 = 1.341

or x = 232.7

For the bottom 9% we have to find the value x for which

P(X<x) = 0.09

Converted to the standard normal distribution this corresponds to

P(Z<(x-178.1)/40.7) = 0.09

Then: (x-178.1)/40.7 = -1.341

or x = 123.5

So the 2 levels that separate the top 9% and the bottom 9% are 232.7 and 123.5.

Answer 2

the levels are 123.5 and 232.7 mg/mL

Explanation:

The inverse normal function is used to find limit values from an area (probability).

Your calculator can show these values to you. (see attached)

The top 9% are above 232.7 mg/mL.

The bottom 9% are below 123.5 mg/mL.

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A suitable calculator can evaluate the inverse normal function to find the x-value corresponding to some area, mean, and standard deviation. For the level that separates the bottom 9% from the rest of the area, we use an area value of 0.09. The mean and standard deviation values are those provided in the problem statement.

For the level that separates the top 9% from the rest of the area, we use an area value of 1 -0.09 = 0.91.

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The second attachment shows an online calculator that can give the limits on the middle 0.91 -0.09 = 0.82 of the area under the probability curve.

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Additional comment

An app version of an approximation to a TI-Nspire calculator is shown in the first attachment. The invNorm function is found at the Distr key (2nd VARS).