Using Normal Distribution tables
By Anonymous on Saturday, December 16, 2000 - 08:05 pm :

I am have trouble evaluating the following:


P(Z<a)=0.7367,

So, Φ(-a)=0.7367

But, from the tables of Normal Distribution, Φ(0.63)=0.7357 and Φ(0.64)=0.7389

In my book they got a=-0.633,

I don't know how they got a from the above data?
Thank you for your help.

By Anonymous on Saturday, December 16, 2000 - 11:28 pm :

linear interpolation

By Kerwin Hui (Kwkh2) on Sunday, December 17, 2000 - 09:25 pm :
Linear interpolation is an approximation in which we assume the function behaves linearly in a small region.

In this example, we have

Φ(0.63)=0.7357, Φ(0.64)=0.7389

So P(Z<a)=0.7367 gives, by linear interpolation

a=0.63+0.01(0.7367-0.7357)/(0.7389-0.7357)=0.633

Kerwin

By Anonymous on Monday, December 18, 2000 - 12:52 pm :

Kerwin, is there is a general form if the method of Linear Interpolation you gave? What if the value of z varied by 2 instead of 1, ie 0.63 and 0.65? instead of only varying by 0.01 in this case? Do you see what I mean?

By Kerwin Hui (Kwkh2) on Monday, December 18, 2000 - 07:52 pm :

It sounds like you would like the formula

f(a)=x, f(b)=y
then f(c)=x+(y-x)(c-a)/(b-a)

although I seriously recommend NOT thinking about linear interpolation by the formula. Instead, look at the idea of joining a straight line betweeen the two given points.

Kerwin

By Anonymous on Tuesday, December 19, 2000 - 03:04 pm :

Thanks.

Can you extend your help to explaining the idea of the straight line? Thanks.

By The Editor :

The tables give us values of the function y=Phi(x) for lots of values of x. If you imagine these plotted as coordinates on a graph, we then have to try and work out what happens inbetween. Since the points are pretty close together, it will not be too inaccurate to assume that the line joining them is straight: we know it is actually curved, but there's not too much difference. You may remember that a straight line graph is linear ; hence the term linear interpolation.

Look at the diagram. The two points we know (in your case, (0.63,Phi(0.63)) and (0.64,Phi(0.64)) are joined by a straight red line. You wanted to find the value of c, given Phi(c).Linear interpolation

If we know what fraction of the way along the straight line we have gone, we can work out what fraction of the distance (b-a) to add to a. That fraction is

begin{displaymath}frac{f(c)-f(a)}{f(b)-f(a)}end{displaymath}



Hence the formula works out to be
begin{displaymath}a + Bigl(frac{f(c)-f(a)}{f(b)-f(a)}Bigr)(b-a)end{displaymath}