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Some Approximations Are Better Than Others

Examine the shape of the binomial histogram for n=20 and p=.10 in Figure  4.9.


  
Figure 4.9: Histogram of Probabilities for Binomial with n=20 and p=.10
\begin{figure}
\begin{center}
\epsfig{file=Bhist201.ps, height=4in, width=3in, angle=-90}\end{center}\end{figure}

Since there is a noticeable skewness to the right, we will get a poor approximation of areas if we replace the rectangles by a normal curve. If the value of p were .90 instead of .10, the binomial histogram would be skewed left. This is typical behavior of binomial histograms whenever p is either too close to 0 or too close to 1. When is it 'safe' to use the normal curve to approximate binomial probabilities? A convenient rule of thumb  is as follows:

\fbox{ \parbox{5.5in}{
\vspace*{1ex}
The normal curve with gives reasonably good...
...omial probabilites whenever both $np>5$\space and $n(1-p)>5$ .
\vspace*{1ex}
} }

The reader is reminded that normal curve approximations, no matter how close, are still approximations. The binomial formula ( 4.3) should be used to calculate exact probabilities whenever possible.




2003-09-08