7. The Central Limit Theorem
7.1 Using the Normal Distribution to Approximate the Binomial Distribution
Recall the definitions: = probability of success,
= probability of failure and
= sample size. When
and
then the normal distribution is very close, numerically, to the binomial distribution.
Using the histogram way of drawing the binomial distribution, a good fit looks like that shown in Figure 7.1.
![](https://openpress.usask.ca/app/uploads/sites/76/2020/01/Chp-7-Figure-2--300x180.jpg)
![Rendered by QuickLaTeX.com \mu = np](https://openpress.usask.ca/app/uploads/quicklatex/quicklatex.com-58c171550e582967452d51ab742f6b5e_l3.png)
![Rendered by QuickLaTeX.com \sigma = \sqrt{npq}](https://openpress.usask.ca/app/uploads/quicklatex/quicklatex.com-058886232993e0c8261c539b1758bfe7_l3.png)
![Rendered by QuickLaTeX.com np \geq 5](https://openpress.usask.ca/app/uploads/quicklatex/quicklatex.com-6688f79929d633f12ae5b97a26f03163_l3.png)
![Rendered by QuickLaTeX.com nq \geq 5](https://openpress.usask.ca/app/uploads/quicklatex/quicklatex.com-5e02b22fa142bd0590b267459f785034_l3.png)
A couple of things to note about this approximation:
- Although the values of the normal and the binomial distributions match well at
equal to integer values when
and
, the areas match not as well. A “correction for continuity” can be used to better make the areas match but we won’t be worrying about such fine details in our studies.
- We will use the normal approximation to the binomial make inferences on proportions. In that case
, the probability of success will represent a proportion in a population.