00:01
For this exercise, we are told that the random variable x is normally distributed with a mean of 50 and a variance of 64.
00:12
Now, for a, we were asked to find the probability that x is greater than 60.
00:18
So we want the probability, x is greater than 60.
00:25
This can also be re -expressed as 1 minus, the probability that x is no more than 60.
00:34
Now, if we want to use the standard normal table, so you solve this, we can standardize.
00:39
It, standardize it according to this mapping.
00:49
Also we need to calculate the standard deviation in this case.
00:52
It's the square root of 64, which is 8.
01:03
So back to the question, we have the probability that z is less than equal to 60 minus the mean over the standard deviation, and this is equal to 1 minus the probability that z is less than equal to 1 .25.
01:26
So now if we go to a standard normal table to look at the same, we'll be a standard normal table to look at probability.
01:31
We want to go to 1 .25, so that corresponds to this probability of 0 .8944.
01:44
And this comes out to 0 .1056.
01:50
So the probability that x is greater than 60 is 0 .106.
01:57
And for b, we want to find the probability that x is greater than 35 and less than 62.
02:14
This can be re -expressed like this.
02:25
For this one, let's use software to solve this.
02:27
We can do this in excel.
02:30
So if we use software, we do not have to spend the time to standardize it.
02:36
So there's two terms to solve probability that x is less than 62 minus the probability that it's less than or equal to 35.
02:45
So in excel, we select a cell and start with equals because we're doing a calculation.
02:50
We want the norm .d .s .t distribution or function.
02:54
So that's the first one highlighted in blue here.
02:57
So we select that.
02:59
And we enter for the first term, it's 62.
03:03
And then we enter the mean and standard deviation of our distribution.
03:07
That's 50 and 8.
03:10
And then for the cumulative argument, we enter true because we want the probability that x is less than or equal to 62.
03:21
And now for the second term, we use the same function, and we enter 35, and the same parameters for our distribution, 50 and 8.
03:32
And again, cumulative is set to true.
03:38
And we get 0 .9028.
03:50
Next for part c, we want to find the probability that x is less than 55.
03:59
Let's use excel again for this one.
04:03
So equals the norm .dd function, 55, and then enter the same distribution parameters, and we get .7 -340.
04:26
And then for d it says the probability is 0 .2 that x is greater than what number.
04:31
So the probability that x is greater than some number is 0 .2.
04:41
And so what is this number? and the equivalent statement is that the probability that x is less than or equal to this number is 1 minus 0 .2 or 0 .8.
04:59
If the probability that it's bigger than x is 0 .2, then the probability that it's not bigger than x must be 0 .8.
05:07
And the reason why we've done this is because we want to to express this as a cumulative probability, the probability that x is less than or equal to something.
05:16
And the reason why we want to do that is because our software has a formula that takes a cumulative probability and then returns the value in the distribution associated with that cumulative probability.
05:35
So what i mean is if this is normal distribution and we enter some cumulative probability, that is probability of being less than this number.
05:56
So in our situation, we want the probability of x being bigger than this number to be 0 .2...