S3 June 2009 2(a)

Suppose the height of an orchid is the random variable X, the mean is m, and standard deviation is s.

We are told that the height of the orchids are normally distributed, so if Z is the standard normal distribution (Z=(X-m)/s) then we are trying to find a number z such that P(-z<Z<z)=0.95. Draw a picture of the bell curve to see that P(-z<Z<z)=0.95 => P(0<Z<z)=0.475 => P(Z<z)=0.975. Look this up on the table to find z=1.96.

Now we just need to shift the interval [-z,z] into the realm of orchid heights (rather than the standard normal distribution). If [g,h] is the range of orchid heights such that 95% of the heights are in the interval [g,h], we can find g and h using g=m-sz and h=m+sz.

In this case, g=20.1-0.5x1.96 and h=20.1+0.5x1.96, so the range is [19.1,21.1] to 1 decimal place.