Question 515704: This is what my text says the properties are, but I am still confused.
The properties of the normal distribution can be summarized as follows:
The normal distribution curve is bell shaped.
The mean, median, and mode are equal. They are located at the center of the distribution.
The normal distribution curve has only one mode (unimodal).
The normal curve is symmetrical about the mean.
The normal curve is continuous.
The normal curve never touches the x-axis. As it extends further in either direction it gets closer to the x-axis but never meets the x-axis.
The total area under the curve is 1.00, or 100%.
68% of the area under the curve lies within one standard deviation of the mean, 95% of the area under the curve lies within two standard deviations of the mean, and 99.7% of the area lies within three standard deviations of the mean.
Here is the question now:
Given a binomial distribution with n = 84 and p = 0.94, would the normal distribution provide a reasonable approximation? Why or why not?
Answer by drcole(72)   (Show Source):
You can put this solution on YOUR website! In general, you can approximate a binomial distribution with a normal distribution with the same mean and standard deviation if the expected number of successes and failures are both at least 5 (some texts use a larger number, but 5 is most common).
If n is the number of experiments and p is the probability of success with each experiment, then np is the expected number of successes after n experiments. In this case:
np = (84)(0.94) = 78.96 successes
which is certainly greater than or equal to 5, so our expected number of successes is high enough to use the normal approximation. If p is the probability of success, then q = 1 - p is the probability of failure. For this problem,
q = 1 - p = 1 - 0.94 = 0.06
which is quite small. The expected number of failures after n experiments would be nq, which in this case gives us:
nq = (84)(0.06) = 5.04 failures
So the expected number of failures is just high enough to use the normal approximation. In other words, because the expected number of successes and failures are both greater than or equal to 5, the binomial distribution with n = 84 and p = 0.94 will be bell-curve shaped enough to justify approximating it with the normal distribution.
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