ck12.org Exercise: Standard Normal Distribution and the Empirical Rule Using the Empirical Rule with a standard normal distribution
ck12.org Exercise: Standard Normal Distribution and the Empirical Rule
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- We're now on problem number 4 from the Normal Distribution
- chapter from ck12.org's FlexBook on AP Statistics.
- You can go to their site to download it.
- It's all for free.
- So problem number 4, and it's, at least in my mind,
- pretty good practice.
- For normal or a standard normal distribution, for a standard
- normal distribution, place the following in order from
- the smallest to largest.
- So let's see, percentage of data below 1, negative 1.
- OK, let's draw our standard normal distribution.
- So a standard normal distribution is one where the
- mean is-- sorry, I drew the standard deviation-- is one
- where the mean, mu for mean, is where the mean is equal
- to 0, and the standard deviation is equal to 1.
- So let me draw that standard normal distribution.
- So let me draw the axis right like that.
- Let me see if I can draw a nice-looking bell curve.
- There's a bell curve right there.
- You get the idea.
- And this is a standard normal distribution, so the mean,
- or you can kind of do the center point right here.
- It's not skewed.
- The mean is going to be 0 right there, and the
- standard deviation is 1.
- So if we go one standard deviation to the right,
- that is going to be 1.
- If you go two standard deviations, it's going to be
- 2, three standard deviations 3, just like that.
- One standard deviation to the left is going to be minus 1.
- Two standard deviations to the left will be minus 2,
- and so on and so forth.
- Minus 3 will be three standard deviations to the left because
- the standard deviation is 1.
- So let's see if we can answer this question.
- So what's the percentage of data below 1?
- So the percentage-- Part a, that's this stuff right here,
- so everything below 1.
- So it's all of-- well, not just that little center portion.
- It keeps going.
- Everything below 1, right?
- Percentage data below 1.
- So this is another situation where we should use
- the empirical rule.
- It never hurts to get more practice.
- Empirical rule.
- Or maybe the better way to remember the empirical rule is
- just the 68, 95, 99.7 rule.
- And I call that a better way because it essentially
- gives you the rule.
- These are just the numbers that you have to
- essentially memorize.
- If you have a calculator or a normal distribution table,
- you don't have to do this.
- But sometimes in class, people want you to estimate
- percentages, and so it's good to do.
- You know, you can impress people if you can do
- this in your head.
- So let's see if we can use the empirical rule to
- answer this question.
- The area under the bell curve all the way up to 1, or
- everything to the left of 1.
- So the empirical rule tells us that this middle area between
- one standard deviation to the left and one standard
- deviation to the right, that right there is 68%.
- We saw that in the previous video as well.
- That's what the empirical rule tells us.
- Now of that 68%, we saw in the last video that everything else
- combined, it all has that up to 1 or to 100%, that this
- left-hand tail-- let me draw it a little bit-- this part right
- here, plus this part right here, has to add up-- when
- you add it to 68-- has to add up to 1 or to 100%.
- So those two combined are 32%.
- 32 plus 68 is 100.
- Now, this is symmetrical.
- These two things are the exact same, so if they add up to 32,
- this right here is 16% and this right here is 16%.
- Now the question, they want us to know the area of
- everything-- let me do it in a new color-- everything
- less than 1, right?
- The percentage of data below 1, so everything to the
- left of this point.
- So it's the 68%, it's right there, so it's 68%, which is
- this middle area within one standard deviation, plus this
- left branch right there.
- So it's 68 plus 16%, which is what?
- That's equal to 84%.
- So this Part a is 84%.
- They're going to want us to put this in order eventually,
- but it's good to just solve essentially the hard part.
- Once we know the numbers, ordering is pretty easy.
- Part b.
- The percentage of data below minus 1.
- So minus 1 is right there.
- So they really just want us to figure out this area right
- here, the percentage of data below minus 1.
- Well, that's going to be 16%.
- We just figured that out.
- And you could have already known, just without even
- knowing the empirical, just looking at a normal
- distribution, that this entire area, that Part b is a subset
- of Part a, so it's going to be a smaller number.
- So if you just have to order things, you could have made
- that intuition, but it's good to do practice with
- the empirical rule.
- Now, Part c, they want to know what's the mean?
- Well, that's the easiest thing.
- The mean of a standard normal distribution, by definition,
- is 0, so number c is 0.
- d, the standard deviation.
- Well, by definition, the standard deviation for
- the standard normal distribution is 1.
- So this is 1 right here.
- This is easier than I thought it would be.
- Part e.
- The percentage of data above 2.
- So they want the percentage of data above 2.
- So we know from the 68, 95, 99.7 rule that if we want to
- know how much data is within two standard deviations-- so
- let me do it in and new color.
- So if we're looking from this-- let me do it in
- a more vibrant color.
- Oh, green.
- If we're looking from this point to this point, so
- it's within two standard deviations, right?
- The standard deviation here is 1.
- If we're looking within two standard deviations, that whole
- area right there by the empirical rule is 95% within
- two standard deviations.
- This is 95%, which tells us that everything else combined--
- so if you take this yellow portion right here and this
- yellow portion right here, so everything beyond two standard
- deviations in either direction, that has to be the remainder.
- So, you know, everything in the middle is 95 within two
- standard deviations, so that has to be 5% If you add that
- tail and that tail together, everything to the left and
- right of two standard deviations.
- Well, I've made the argument before.
- Everything is symmetrical.
- This and this are equal, so this right here is 2 and a half
- percent, and this right here is also 2 and a half percent.
- So they're asking us the percentage of data above 2.
- That's this tail, just this tail right here.
- The percentage of data more than two standard deviations
- away from the mean, so that's 2 and a half percent.
- I'll do it in a darker colors 2 and a half percent.
- Now, they're asking us, place the following in order
- from smallest to largest.
- So there's a little bit of ambiguity here, because if
- they're saying the percentage of data below 1, do they want
- us to say, well, it's 84%.
- So should we consider the answer to Part a 84, or should
- we consider-- if they said the fraction of data below
- 1, I would say 0.84.
- So it depends on how they want to interpret it.
- Same way here.
- The percentage of data below minus 1, I could
- say the answer is 16.
- 16 is the percentage below minus 1, but the actual number,
- if I said the fraction of data below minus 1, I would say
- 0.16, so this actually would be very different in
- how we order it.
- Similarly, if someone asked me the percentage, I would say,
- oh, that's 2.5, but the actual number is 0.025.
- That's the actual fraction or the actual decimal.
- So, I mean, this is just ordering numbers so I shouldn't
- fixate on this too much.
- But let's just say that they're going with the decimal.
- So if we wanted to do it that way, if they wanted to do it
- from smallest to largest, the smallest number we have
- here is c, right?
- That's 0.
- And then the next smallest is e, which is 0.025.
- Then the next smallest is b, which is 0.16.
- And then the next one after that is a, which is 0.84, and
- then the largest would be the standard deviation d.
- So the answers is cbad.
- And obviously, the order would be different if the answer to
- this, instead of saying it is 0.084, you said it was 84,
- because they're asking for the percentage.
- So a little bit of ambiguity.
- If you had a question like this on the exam, I would clarify
- that with the teacher.
- But hopefully, you found this useful.
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