00:02
Here we are told that x is normally distributed with a mean of 304 .6 and a standard deviation of 13 .2.
00:11
I just want to point out that the font on the website came out a little funny, so i'm assuming that it's saying sigma equals 13 .2.
00:21
If it says sigma squared equals 13 .2, it means the variance is 13 .2, in which case you will have to take the square root of it to express it as sigma, as the standard.
00:33
And of course that would modify your answer a little bit.
00:39
But i believe that it's saying sigma equals 13 .2.
00:43
That is the standard deviation.
00:46
So we are asked to find the probability that x is less than 315 .9, given that we know that it is larger than 288 .4.
01:09
So graphically, it looks like this.
01:11
If this is the distribution for x, we have our mean of 304 .6.
01:19
Everything to the right of 288 .4 is the distribution for is the probability that x is greater than 288 .4.
01:31
We want the probability that x is less than 315 .9 given that it's in the blue region.
01:43
So we know it's somewhere in here to the right of 288 .4.
01:48
Given that what is the probability that it is somewhere in this region to the left of 350 .9? the answer is it's the area of the red -shaded region divided by the area of the blue -shaded region.
02:07
Mathematically, this can be stated like this.
02:10
This conditional probability is equal to probability that x is less than 350 .9 intersected with x is greater than 288 .4.
02:27
This is from probability theory divided by the probability that x is greater than 288 .4.
02:40
This is from this probability axiom.
03:02
And so to be bigger than 288 .4 but less than 315 .9, this probability can be expressed like this.
03:30
And so in the numerator, we can re -express this probability as the probability that x is less than 315 .9, minus the probability that x is at most 288 .4.
03:48
And in the denominator, we can re -express it as 1 minus the probability that x is at most 288 .4...