1
5.2 Normal Approximation to the
Binomial
Purpose:
To develop a better understanding of why and how well the normal
approximation to the binomial distribution works.
Reading
Assignment:
Read through Section 5.4.
Problem
Description:
For various combinations of n, the number of trials,
and p, the probability of success, stored Minitab commands will be executed
which do the following:
a)
Compute mu, sigma, mu+3(sigma), and mu-3(sigma);
Compute P(x<=k), where k is mu-sigma rounded to the nearest integer, using
both the binomial distribution and the normal approximation to the binomial
distribution; Compute P(x<=k), where k is mu+sigma rounded to the nearest
integer, using both the binomial distribution and the normal approximation to
the binomial distribution;
b)
Graph the corresponding normal and binomial
distributions.
Your job
in each case
is to
check
if the approximation
applies (i.e. is 0 <= mu - 3sigma, and is mu + 3 sigma <= n?), see
how well
the exact and approximate probabilities compare, and see
how well
the
functions match up on the graph.
In Part II of Lab 4.2, we generated plots of various binomial distributions,
first fixing n=25 and varying p away from a half, then fixing p=0.25 and
varying n. We will do the same in this lab, but in each case for each
binomial distribution we will overlay a plot of the corresponding normal
distribution for comparison. We will also use the
cdf
command to compute P(x
<= mu-sigma) and P(x <= mu+sigma) both exactly (using the binomial
distribution) and approximately (using the normal approximation) to see how
close the approximation comes. The Minitab "program" to do all this is a bit
too long and involved to merit digesting here. Instead, the necessary
commands have been stored in the file "nab.MTB", (where "nab" stands for
Normal Approximation to Binomial).
Step 1: Move a file on your disk.
The file "nab.MTB" is on your disk,
in the folder "STT 264 Data". For the file to be used, you must first move it
out of the folder. Thus, open your diskette to a window, then open the folder
"STT 264 Data" to a window, then drag the file "nab.MTB" from the folder
window to the diskette window. (You can also find the folder by selecting
"Software for ALL users -> Stats Data -> STT 264 Data" to open "STT 264 Data"
to a window, then drag the file from "STT 264 Data" into your disk window.)
Then close all of the windows you just opened, except the window for your
diskette.
Step 2:
Start Minitab from your diskette ('your diskette' -> Start
Minitab.MTB)
Step 3:
Read the values n=25 and p=0.5 into c1 and c2 as follows.
MTB > read c1-c2
DATA> 25 0.5
DATA> end
1 ROWS READ
(Look at the worksheet to see what is there.)
