STA220H5 Chapter 5.5: (5.5-5.6)
(5.5-5.6)
Using a normal distribution to approximate binomial probabilities
1. After you have determined n and p for the binomial distribution, calculate
the interval
U+-3a=np+-3npq
If the interval lies in the range from 0 to n, the normal distribution will
provide a reasonable approximation to the probabilities of most binomial
events.
2. Express the binomial probability to be approximated in the form P (x<a) or P
(x<b)-P (x<a). For example,
P (x<3)=P (x<2)
P (x>5)=1-P (x<4)
P (7<x<10)=P (x<10)-P (x<6)
3. For each value of interest, a, the correction for continuity is (a+. 5) and the
corresponding standard normal z value is
Z=(a+. 5)-u/a (see figure 5.26)
4. Sketch the approximating normal distribution and shade the area
corresponding to the probability of the event of interest, as in figure 5.26.
Verify that the rectangles you have included in the shaded area correspond to
the probability you wish to approximate. Use the z value(s) you calculated in
step 3 to find the shaded area with table II or technology. This is the
approximate probability of the binomial event.
Figure 5.26
Approximating binomial probabilities by normal probabilities.
Probability distribution for an exponential random variable x
Probability density function: f (x)=1/0e-x/0 (x>0)
Mean: u=0 standard deviation: a=0
Finding the area A to the right of a number a for an exponential
distribution*
A=P (x>a0=e-a/0
Figure 5.28
The are A to the right of a number a for an exponential distribution
Document Summary
Using a normal distribution to approximate binomial probabilities: after you have determined n and p for the binomial distribution, calculate the interval. P (7
