00:01
So for this problem, based on the context here, i'm going to assume that since the mean value is equal to 26, that's equal to lambda, which is in turn greater than 5, that we are using the normal approximation to the poisson distribution.
00:29
So what i'll do is i'll go through the process of finding the probabilities using the normal approximation, and then just confirm the results afterwards using direct calculation with my software, just to show the accuracy of the normal approximation.
00:44
So to begin to find the probability that x is less than or, oops, i forgot to write x, that x is less than or equal to 14, we can instead find the probability that a z score is less than the z score corresponding to that value, which would be x minus lambda so that would be oh also i'll note i'm going to apply the continuity oh excuse me the continuity correction factor so because this is x less than or equal to 14 i'll translate that to x less than 14 .5 minus lambda over the square root of lambda so finding that said score you have that that's going to be 14 .5 minus 26 divided by the square root of 26 which gives us a z score of negative 2 .26 roughly.
01:46
Now for this problem, i'll show how we would find this probability using a table of values, but for the subsequent problems i'll just be using my software.
01:54
So we know that we're looking for probability to the left of negative 2 .26.
01:59
We want to confirm we're using the right kind of table here.
02:01
We find the negative 2 .2 row, the 0 .06 column, find the intersection of the two, and we find that that probability is going to be 0 .0 .0.
02:12
119.
02:17
Then for part b, i'll note that this is one that we can actually calculate without too much of a headache.
02:25
One moment here.
02:27
Since we have that we can basically, or not basically, we can apply the probability mass function for a poisson exactly here.
02:35
So probability that x equals 16, we can calculate just as lambda to the power of 16 times e to the power of negative lambda divided by 16 factorial.
02:47
So, 26 to the power of 16 times e to the power of negative 26, divided by 16 factorial, gives us a probability of 0 .0106.
03:02
So this one, like i said, we don't need to apply the approximation.
03:06
It's pretty straightforward to calculate that.
03:09
Though what i'll do is that's the exact value.
03:12
We can also find the approximate value, by finding the probability of being basically in an interval of width 0 .5 about that x value.
03:26
So probability of z being between the z scores corresponding to 15 .5 and 16 .5.
03:34
I'll go into less detail with the calculation here, basically just showing the results.
03:39
All right, so these are the corresponding z scores, the lower and upper bounds, and we'd find that using the normal approximation, our results, would be about 0 .1115.
03:50
Then moving on to part c, probability of x greater than 21.
03:58
We can find by, again, finding our corresponding z score...