00:01
Hey there, welcome to numerate.
00:04
We are given a unimodal and symmetric distribution here, in which we have our population mean equaling to 35 .9 points and a population standard deviation that equals 11 .6 points.
00:22
So we're basically looking at the pollution index for a sample of cities here.
00:28
So starting with our setup here, we know this follows our empirical rule.
00:36
So we can use our empirical rule to estimate our following percentages here.
00:43
So we have one standard deviation, two standard deviations, and three standard deviations.
00:57
So with this here, one standard deviation away gives us a percentage of 68 .27%.
01:09
2 standard deviations, 95 .45%, and 3 standard deviations is 99 .73%.
01:20
So let's start with part a here and see how far the values are from the mean.
01:28
So for part a, we have the probability between 12 .7 and 59 .1.
01:42
So we can just take one of these limits here and calculate our z score to see how many standard deviations it is away from the mean.
01:52
So let's take the upper one here and then subtract it to our mean, 35 .9, divided by our standard deviation, 11 .6.
02:05
Therefore, this will give us a z score here.
02:10
That equals.
02:12
So within two, we know that it is 68 .2.
02:17
Percent for one and a two is 95 .45 % here for part a.
02:27
All right.
02:29
Let's try part b.
02:31
We're doing the same thing here, trying to identify this z score first.
02:35
So what we have is between 24 .3 and 47 .5...