00:01
There is a normal distribution here and we have to get the z scores.
00:05
For which the 90 % of the area lies between negative z and z.
00:10
So the normal distributions are symmetrical with respect to mean value.
00:16
So for the z scores, the mean value is zero and the standard division is one.
00:21
So let me just find part a.
00:24
So if i draw in bell shape here, this is the new value and there is some z value here.
00:31
Here and symmetrical and negative z here.
00:35
So the area between those two values that is given in the question as 90%.
00:40
So the 90 % means this is 0 .9.
00:44
So the probability of, let's say, x is the random variable here.
00:52
So x is between z and negative z.
00:55
So the probability is 0 .9.
00:58
Actually, so what we have to do, we have to just find the z here.
01:04
Here.
01:05
So since this is symmetrical, we know that the area of this part is 0 .45 and the area of this part is 0 .5 .45.
01:13
So the probability of, and this part has the probability of 0 .5 and this part has the 0 .45.
01:20
If i add this together, we will get 0 .95.
01:23
So the probability of x is less than z, which should be equal to 0 .95.
01:29
In order to get the z, we have to use the inverse norm.
01:34
And let me just show you from the calculator.
01:37
Just press second and distribution.
01:39
There is inverse norm.
01:41
And the area is 0 .95 comma.
01:45
The mu value is zero because this is standard normal curve and this standard division is 1.
01:51
So we got the z value as, which is 1 .645.
01:58
And what about negative z? that is negative 1 .645.
02:01
So the area is between the z values 1 .6445 and negative 1 .6445.
02:10
And for b, what about the z score for which the 80 % of the distribution lies? no, actually, sorry, the second one is find two scores for which the 90 % of the distribution area lies between negative z and z.
02:26
So this is very similar to the previous case.
02:30
This is mu and there's some value z here and negative z here so this shaded region in this case is given as 98 so that means this is equal to 0 .98 so the area of this region sorry about that the area of this region is 0 .49 and this is 0 .49 and the area of this part which is the half of the normal distribution 0 .5 and this area is 0 .49.
03:00
If i just add them up i will get 99.
03:04
So the probability of x is less than z, which is 0 .99...