ck12.org Normal Distribution Problems: Empirical Rule Using the empirical rule (or 68-95-99.7 rule) to estimate probabilities for normal distributions
ck12.org Normal Distribution Problems: Empirical Rule
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- Let's do another problem from the normal distribution
- section of ck12.org's AP statistics book.
- And I'm using theirs because it's Open Source and it's
- actually quite a good book.
- The problems are, I think, good practice for us.
- So let's see, number 3.
- You could go to their site and I think you
- can download the book.
- Assume that the mean eight of 1 year old girls in the U.S. is a
- normally distributed-- or is normally distributed with the
- mean of about 9.5 grams.
- That's got to be kilograms.
- I have a 10 month old son and he weighs about 20
- pounds which is about 9 kilograms not 9.5 grams.
- 9.5 grams is nothing.
- This would be talking about like mice or something.
- This has got to be kilograms.
- But anyway, it's about 9.5 kilograms with a
- standard deviation of approximately 1.1 grams.
- So the mean is equal to 9.5 kilograms I'm assuming and
- the standard deviation is equal to 1.1 grams.
- Without using a calculator-- so that's an interesting clue--
- estimate the percentage of 1 year old girls in the U.S. that
- meet the following conditions.
- So when they say that without a calculator estimate that's a
- big clue or a big giveaway that we're supposed to use
- the empirical rule.
- Empirical rule sometimes called the 68-95-99.7 rule.
- And if you remember this is the name of the rule
- you've essentially remembered the rule.
- What that tells us that if we have a normal distribution--
- I'll do a bit of a review here before we jump
- into this problem.
- If we have a normal distribution-- let me draw
- a normal distribution.
- It looks like that.
- That's my normal distribution.
- I didn't draw it perfectly but you get the idea.
- It should be symmetrical.
- This is our mean right there.
- That's our mean.
- If we go one standard deviation above the mean and one standard
- deviation below the mean, so this is our mean plus
- one standard deviation.
- This is our mean minus one standard deviation.
- The probability of finding a result if we're dealing with a
- perfect normal distribution that's between one standard
- deviation below the mean and one standard deviation above
- the mean-- that would be this area-- and it would be,
- you could guess, 68%.
- 68% chance you're going to get something within one standard
- deviation of the mean.
- Either a standard deviation below or above or
- anywhere in between.
- Now, if we're talking about two standard deviations around the
- mean-- so if we go down another standard deviation, we go down
- another standard deviation in that direction and another
- standard deviation above the mean-- and we were to ask
- ourselves what's the probability of finding
- something within those two or within that range, then it's,
- you could guess it, 95%.
- And that includes this middle area right here.
- So the 68% is a subset of that 95%.
- And I think you know where this is going.
- If we go three standard deviations below the mean and
- above the mean, the empirical rule or the 68-95-99.7 rule
- tells us that there is a 99.7% chance of finding a result in a
- normal distribution that is within three standard
- deviations of the mean.
- So above three standard deviations below the mean
- and below three standard deviation above the mean.
- That's what the empirical rule tells us.
- Now let's see if we can apply it to this problem.
- So they gave us the mean and the standard deviation.
- Let me draw that out.
- Let me draw my axis first as best as I can.
- That's my axis.
- Let me draw my bell curve.
- That's about as good as a bell curve as you can expect
- a freehand drawer to do.
- And the mean here is 9.-- and this should be symmetric.
- This height should be the same as that height there.
- I think you get the idea.
- I'm not a computer.
- 9.5 is the mean.
- I won't write the units.
- It's all in kilograms.
- One standard deviation above the mean we should add 1.1 to
- that because they told us the standard deviation is 1.1.
- That's going to be 10.6.
- If we go-- let me just draw a little dotted line there-- 1
- standard deviation below the mean we're going it subtract
- 1.1 from 9.5 and so that would be 8.4.
- If we go two standard deviations above the mean
- we would add another standard deviation here.
- We went one standard deviations, two
- standard deviations.
- That would get us to 11.7.
- And if we were to go three standard deviations
- we'd add 1.1 again.
- That would get us to 12.8.
- Doing it on the other side, one standard deviation
- below the mean is 8.4.
- Two standard deviations below the mean-- subtract 1.1
- again-- would be 7.3.
- And then three standard deviations below the mean--
- which we'd write there-- would be 6.2 kilograms.
- So that's our set up for the problem.
- So what's the probability that we would find a one year old
- girl in the U.S. that weighs less than 8.4 kilograms.
- Or maybe I should say whose mass is less
- than 8.4 kilograms.
- So if we look here, the probability of finding a baby
- or a female baby who is one year old with a mass or a
- weight of less than 8.4 kilograms, that's this
- area right here.
- I said mass because kilograms is actually a unit of mass.
- Most people use it as weight as well.
- So that's that area right there.
- So how can we figure out that area under this
- normal distribution using the empirical rule?
- Well, we know what this area is.
- We know what this area between minus one standard deviation
- and plus one standard deviation is.
- We know that is 68%.
- And if that's 68% then that means in the parts that
- aren't in that middle region you have 32%.
- Because the area under the entire normal distribution is
- 100 or 100% or 1, depending on how you want to think about it.
- Because you can't have-- well, all of the possibilities
- combined can only add up to 1.
- You can't have it more than 100% there.
- So if you add up this leg and this leg-- so this plus that
- leg is going to be the remainder.
- So 100 minus 68, that's 32.
- 32% is if you add up this left leg and this
- right leg over here.
- And this is a perfect normal distribution.
- They told us it's normally distributed.
- So it's going to be perfectly symmetrical.
- So if this side and that side add up to 32 but they're both
- symmetrical, meaning they have the exact same area, then this
- side right here-- I'll do it in pink-- this side right here--
- it ended up looking more like purple-- would be 16%.
- And this side right here would be 16%.
- So your probability of getting a result more than one standard
- deviation above the mean-- so that's this right hand
- side, would be 16%.
- Or the probability of having a result less than one standard
- deviation below that mean, that's this right here, 16%.
- So they want to know the probability of having a
- baby at one years old less than 8.4 kilograms.
- Less than 8.4 kilograms is this area right here.
- And that's 16%.
- So that's 16% for part a.
- Let's do part b: between 7.3 and 11.7 point seven kilograms.
- So between 7.3-- that's right there.
- That's two standard deviations below the mean-- and 11.7, one,
- two standard deviations above the mean.
- So there's essentially asking us what's the probability of
- getting a result within two standard deviations
- of the mean, right?
- This is the mean right here.
- This is two standard deviations below.
- This is two standard deviations above.
- Well that's pretty straightforward.
- The empirical rule tells us between two standard deviations
- you have a 95% chance of getting a result that is within
- two standard deviations.
- So the empirical rule just gives us that answer.
- And then finally, part c: the probability of having a one
- year old U.S. a baby girl more than 12.8 kilograms.
- So 12.8 kilograms is three standard deviations
- above the mean.
- So we want to know the probability of having a result
- more than three deviations above the mean.
- So that is this area way out there that I drew in orange.
- Maybe I should do it in a different color to
- really contrast it.
- So it's this long tail out here, this little small area.
- So what is that probability?
- So let's turn back to our empirical rule.
- Well we know the probability-- we know this area.
- We know the area between minus three standard deviations and
- plus three standard deviations.
- We know this-- since this is last the last problem I can
- color the whole thing in-- we know this area right here
- between minus 3 and plus 3, that is it 99.7%.
- The bulk of the results fall under there.
- I mean, almost all of them.
- So what do we have left over for the two tails?
- Remember there are two tails.
- This is one of them.
- Then you have the results that are less than three standard
- deviations below the mean.
- This tail right there.
- So that tells us that this, less than three standard
- deviations below the mean and more than three standard
- deviations above the mean combined have to be the rest.
- Well the rest, there's only 0.3% percent for the rest.
- And these two things are symmetrical.
- They're going to be equal.
- So this right here has to be half of this or 0.15% and
- this right here is going to be 0.15%.
- So the probability of having a one year old baby girl in the
- U.S. that is more than 12.8 kilograms if you assume a
- perfectly normal distribution is the area under this curve,
- the area that is more than three standard deviations
- above the mean.
- And that is 0.15%.
- Anyway, I hope you found that useful.
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