## Simultaneous Equations Problems Involving the Normal Distribution

Sometimes it happens that we know a population is normally distributed but we don't have a full set of data and so cannot find the mean and standard deviation from a set of readings. However, given JUST TWO suitable statistics we can find the mean and standard deviation of a list. This is because there are two constants we must find, and any set of two unknowns can be found with two linked – simultaneous – equations.

The statistics we must be given can be any two of the form which is equivalent to for a continuous distribution. For each of these we go to the normal distribution tables and look up the corresponding value of z. We have then two sets of values (x,z) which we substitute into the equation then solve the simultaneous equations.

Example: The duration of the pregnancy of a certain breed of cow is normally distributed with mean days and standard deviation days. Only 2.5% of all pregnancies are shorter than 235 days and 15% are longer than 286 days. Find and 2.5% of all pregnancies are shorter than 235 days so look up 1-0.025=0.975 in the normal distribution tables. Our tables have 0.5 subtracted from every entry so we look up the z value corresponding to The first equation is (1)

15% are longer than 286 days Subtract 0.5 as before to obtain 0.35 and look up the value of corresponding to this probability in the tables, obtaining – this requires us to interpolate. The second equation is (2)

We solve the simultaneous equations (1) (2)

(2)-(1) From (1)  