00:01
Hey there, welcome to numerate.
00:03
We are asked to find a probability that an 18 -year -old man is between 67 and 71 inches tall, given that we're given the population standard deviation and mean here.
00:18
So let's write them down.
00:20
The mean that we're given here is equal to 68, and our standard deviation equals to 3.
00:36
All right, so we have 68 and 3 as our parameters here.
00:49
Now we are going to be using these two values here to solve for our probability.
01:01
So our first part here, part one, asks us to find a probability that x is between 67 and 71 inches tall.
01:16
So in order to find this probability here, what we would have to find and do is to expand this probability.
01:28
So we can expand this out into the probability where x is less than 71 minus the probability where x is less than 67.
01:49
So we're going to set up z -score equations for both of these.
01:53
We have our 71 minus 68 which is 3 3 divided by 3 equals 1 so with a z score of 1 we get a probability of around 0 .84134 and this is going to be subtracted to our other probability here where we have the z score of 70 67 minus 68 divided by 3 equaling negative 1 3rd to negative 0 .33.
02:28
And for this here, this gives us a c score that equals around for 67 of 0 .36944.
02:51
So we subtract giving us an in -between probability of around 0 .4719.
03:15
0 .4719.
03:19
Alright, so our next one here is part two.
03:30
So part two will be pretty similar here, but now we're giving a sample size of 36.
03:36
So we're still trying to find the probability, where now x mean is between 67 and 71.
03:58
So this basically equals the difference between the two probabilities.
04:05
So it's exactly same as above, but we're dealing with x bar x meet.
04:14
So to find the z scores now, we can do a shortcut method.
04:24
Z score equals our positive 1 from above multiplied by the square root of 36.
04:33
So this here will basically be 6 times 1, which is 6.
04:38
For this c score here, we're going to do the same, negative 0 .33, multiplied by the square root of 36.
04:53
All right, so with this here, we have a c score of 6, which corresponds to a probability of basically 1...