This tutorial talks about the normal distribution approximation of the binomial distribution. Now I know that's a lot of words. But it actually is pretty easy. It's two concepts that you already know about. It's the normal distribution and the binomial distribution.
Now the reason this is interesting is because we can use the normal distribution to approximate a binomial distribution if certain things are true. So if the number of trials is large and the probability of success is near the middle, then you can use the normal distribution to estimate the number of successes. Now when we're looking at this, we're going to say that n is the number of trials, p is the probability success, and q is the probability of failure. And then because success and failure are complements, q is going to be the same as 1 minus p.
So here, when you're wanting to calculate the mean, it's going to be n times p. The variance is going to be n times p times q. And the standard deviation is going to be the square root of the variance, so the square root of n times p times q.
Now for example, if a baseball player is a 0.300 hitter, what percent of the time can he be expected to hit between 20.8 and 39.2 hits in 100 at bats? Now because we're doing a large number of trials and our probability of success is near the middle, then we're going to use the normal distribution to approximate the number of successes. So, first, we need to find out what the mean is. So our mean is n times p.
Now let's go back through the problem to find n, p, and q. So here, this 0.300, how often he hits, is his probability of success. So the q is going to be 1 minus 0.300, which is going to be 0.700. And the n, the number of trials, is 100, how many times he's getting up to bat.
So for the mean, it's going to be n times p. So it's going to be 100 times 0.300, which is 30. Then the variance is going to be n times p times q, so 100 times 0.300 times 0.700. Which when you enter that into your calculator, you get that it's 21. The standard deviation is the square root of the variance. So the square root of 21, which rounds to be 4.6.
So those are the key pieces of information that we're going to use to help to build the normal curve for the situation. So not the normal curve. Sorry. That's going to help to build the curve for the situation.
So here, we've got our curve. And it's not perfect, but it'll work. In the middle is the mean. And then I'm adding on these standard deviation bars up and down.
So we say 30 plus 4.6 is going to get us this first deviation up. And then 30 minus 4.6 is going to get us the next one down. And then we're going to add on another 4.6 to here. And then I'm going to subtract another 4.6 from here.
Now I'm going to stop because these are the numbers the problem was asking about, 20.8 and 39.2. So they want to know, what's the probability of falling between those two bars? So what percent of the time does he do that?
And if you remember your 68, 95, 99.7 rule, 68% of the time he's between the first two bars. 95% of the time, he's between the second two standard deviations, so two standard deviations up to two standard deviations down from the mean. So then that means that 95% of the time, in 100 at bats, this player is going to hit between 20.8 and 39.2 hits. So again, sorry. 95% of the time, he's going to hit between 20.8 and 39.2 hits.
And that's because we knew the mean was 30. We calculated that the standard deviation was 4.6. And then once we drew our curve and we know that we're two standard deviations up from the mean and two standard deviations down from the mean, we thought back to our 68, 95, 99.7 rule and knew that it was 95% of the time between two standard deviations up and two standard deviations down. This has been your tutorial on the normal distribution approximation of the binomial distribution.