We are given that the mean systolic blood pressure of a population (women aged 18 to 24) is `mu=120.3` and the standard deviation of this distribution is `sigma=12.7`. We are also told that the distribution is normal. We are asked to find the interval that contains the middle 90% of...

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We are given that the mean systolic blood pressure of a population (women aged 18 to 24) is `mu=120.3` and the standard deviation of this distribution is `sigma=12.7`. We are also told that the distribution is normal. We are asked to find the interval that contains the middle 90% of women aged 18–24.

A normal distribution is symmetric about the mean. This means that there is reflexive symmetry about the central point. The standard normal distribution has a mean of zero and a standard deviation of 1. Normal distributions can be converted to the standard normal distribution (normalized) by converting the data to z-values.

If 90% of the population is in the interval, 10% lie outside the interval; 5% below the interval and 5% above, by symmetry. In the standard normal distribution, the z-score that corresponds to 5% below is `z~~-1.65`. We can get this value from a standard normal table (you will find that .0500 lies between -1.64 and -1.65) or using some utility. My calculator returns -1.644853626. This number comes from the function that describes the normal or Gaussian distribution.

In a similar fashion, we find the z-score with 95% of the population above it is about 1.65.

We use these values to find the data points in the actual distribution.

If z=-1.65, then `x=z(sigma)+mu~~-1.65(12.7)+120.3=99.3`,

and if z=1.65, then `x=1.65(12.7)+120.3~~141.3`. (This means that a systolic reading of 99.3 lies 1.65 standard deviations below the mean and represents the bottom 5% of the population.)

**The interval for the middle 90% is 99.3<x<141.3**

**Further Reading**