Chapter 1. Normal Approximation to Binomial Distributions
Introduction
You can use the sliders to change both n and p. Click and drag a slider with the mouse. Start by choosing p. The binomial distributions are symmetric for p = 0.5. They become more skewed as p moves away from 0.5. The bars show the binomial probabilities. The vertical gray line marks the mean np. The red curve is the normal density curve with the same mean and standard deviation as the binomial distribution. As you increase n, the binomial probability histogram looks more and more like the normal curve.
The Central Limit Theorem says that as n increases, the binomial distribution with n trials and probability p of success gets closer and closer to a normal distribution. That is, the binomial probability of any event gets closer and closer to the normal probability of the same event. The normal distribution has the same mean μ = np and standard deviation as the binomial distribution.
1.
Suppose you flip a "fair" coin (that is, one with probability 0.5 of coming up either heads or tails) 16 times. According to the applet, the most likely result will be that of the tosses will come up heads. The probability of this outcome, according to the applet, is approximately .
2.
Now increase the number of trials to 50. According to the applet, the most likely result will be that of the tosses will come up heads. Compared to the distribution of results with 16 trials, the distribution with 50 trials resembles the normal curve, because with more possible outcomes, the distribution is .
3.
Set the number of trials to 10, then compare the distribution of results when p = 0.2, 0.5, and 0.7. What are the most likely outcomes (that is, the most likely number of heads you will throw in 10 tosses) in these three cases? State in your own words the relationship between the probability of heads, the number of trials, and the most likely outcome for this number of trials.