00:01
It's estimated that 3 .6 % of the general population will live past their 90th birthday.
00:06
So that's a probability of 0 .036 for any randomly selected person.
00:12
And we consider a graduating class of 759 high school seniors.
00:18
And we want to find the probabilities that certain numbers of them live beyond their 90th birthday.
00:23
So if we define a random variable x as the number of high school seniors who live beyond on their 90th birthday, then x is a binomial random variable with parameters n equals 759 and p equals 0 .036.
00:41
We can use the normal approximation to the binomial with pretty good accuracy if n times p is greater than 5 and n times 1 minus p also greater than than 5.
00:55
So n times p is 759 times 0 .036, which is 27 .324.
01:08
So that's greater than 5.
01:13
And n times 1 minus p.
01:16
1 minus p is greater than p because p is less than half.
01:19
So this is also bigger than 5.
01:24
So if we apply the normal approximation, the normal approximation says says that the number of successes in a binomial distribution is approximately normal with a mean equal to n times p and a standard deviation the square root of n times p times 1 minus p.
01:56
And so if we calculate the mean we get 27 .324 and the standard deviation 5 .132 approximately.
02:14
So now for part we want the probability that 15 or more will live beyond their 90th birthday.
02:19
So we want the probability that x is at least 15, which can be expressed as 1 minus the probability that x is at most 14.
02:30
And because we're using a continuous distribution to model a discrete distribution, we can apply a continuity correction factor and state this as the probability that x is at most 14 .5.
02:45
And now we can solve this using excel.
02:48
We start with an equal sign.
02:50
We first have 1 minus, then we want to use the normal distribution function.
02:54
We select that function, enter 14 .5 for the first argument, then the mean, that's 27 .324, standard deviation 5 .132.
03:05
We enter true for the cumulative argument to get the probability of any number from 0 up to 14 .5.
03:12
Hit enter, and we get .9938 approximately.
03:18
If we're calculating to four decimal places, perhaps we'll use four decimal places on the standard deviation...