Normal distribution
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Introduction to the Normal Distribution
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Normal Distribution Excel Exercise
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ck12.org Normal Distribution Problems: Qualitative sense of normal distributions
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ck12.org Normal Distribution Problems: Empirical Rule
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ck12.org Normal Distribution Problems: z-score
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ck12.org Exercise: Standard Normal Distribution and the Empirical Rule
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Empirical rule
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ck12.org: More Empirical Rule and Z-score practice
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Z scores 1
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Z scores 2
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Z scores 3
ck12.org: More Empirical Rule and Z-score practice More Empirical Rule and Z-score practice
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- Never hurts to get a bit more practice.
- So this is problem number 5 from the Normal Distribution
- chapter from ck12.org's AP Statistics FlexBook.
- So they're saying the 2007 AP Statistics examination scores
- were not normally distributed with a mean of 2.8 and a
- standard deviation of 1.34.
- They cite some College Board stuff here.
- I didn't copy and paste that.
- What is the approximate z-score?
- Remember, z-score is just how many standard deviations
- you are away from the mean.
- What is the approximate z-score that corresponds
- to an exam score of 5?
- So we really just have to figure out-- this is a pretty
- straightforward problem-- we just have to figure out how
- many standard deviations is 5 from the mean?
- Well, you just take 5 minus 2.8, right?
- The mean is 2.8.
- Let me be very clear.
- Mean is 2.8.
- They give us that.
- We didn't even have to calculate it, right?
- So the mean is 2.8, so 5 minus 2.8 is equal to 2.2.
- So we're 2.2 above the mean, and if we want that in terms
- of standard deviations, we just divide by our
- standard deviation.
- Divide by 1.34.
- Divide by 1.34.
- I'll take out the calculator for this.
- So we have 2.2 divided by 1.34 is equal to 1.64.
- This is equal to 1.64, and that's choice c.
- So this was actually very straightforward.
- We just had to see how far away we are from the mean if we get
- a score of 5, which hopefully you will get if you're taking
- the AP Statistics exam after watching these videos, and then
- you divide by the standard deviation to say how many
- standard deviations away from the mean is the score of 5.
- It's 1.64.
- I think the only tricky thing here might have been-- you
- might have been tempted to pick choice e, which says the
- z-score cannot be calculated because the distribution
- is not normal.
- I think the reason why you might have had that temptation
- is because we've been using z-scores within the context
- of a normal distribution.
- But a z-score literally just means how many standard
- deviations you are away from the mean.
- It could apply to any distribution that you can
- calculate a mean and a standard deviation for.
- So e is not the correct answer.
- A z-score can apply to a non-normal distribution, so the
- answer is c, and I guess that's a good point of clarification
- to get out of the way.
- I thought I would do two problems in this video, just
- because that was pretty short.
- So problem number 6: The heights of fifth grade boys in
- the United States is approximately normally
- distributed-- that's good to know-- with a mean height of
- 143.5, so it's a mean of 143.5 centimeters, and a standard
- deviation of about 7.1 centimeters.
- Standard deviation of 7.1 centimeters.
- What is the probability that our randomly chosen fifth
- grade boys would be taller than 157.7 centimeters?
- So let's just draw out this distribution like we've done in
- a bunch of problems so far.
- They're just asking us one question, so you can mark this
- distribution up a good bit.
- Let's say that's our distribution-- and the
- mean here, the mean they told us is 143.5.
- They're asking us taller than 157.7, so we go in
- the upwards directions.
- So one standard deviation above the mean will take us right
- there, and we just have to add 7.1 to this number right here.
- We're going up by 7.1.
- So 143.5 plus 7.1 is what?
- 150.6.
- That's one standard deviation.
- If we were to go another standard deviation,
- we go 7.1 more.
- What's 7.1 plus 150.6?
- It's 157.7, which just happens to be the exact
- number they ask for.
- They're asking for heights, the probability of getting
- a height higher than that.
- So they want to know what's the probability that we fall under
- this area right here, or essentially more than two
- standard deviations from the mean, or above two
- standard deviations.
- We can't count this left tail right there.
- So we can use the empirical rule.
- We can use the empirical rule.
- If we do our standard deviations, to the left that's
- one standard deviation, two standard deviations.
- We know what this whole area is.
- Let me pick a different color.
- So we know what this area is, the area within two
- standard deviations.
- The empirical rule tells us.
- Or even better, the 68-95-99.7 rule tells us that this area,
- because it's within two standard deviations, is 95%, or
- 0.95, or it's 95% of the area under the normal distribution,
- which tells us that what's left over, this tail that we care
- about and this left tail right here, has to make
- up the rest of it, or 5%.
- So those two combined have to be 5%, and these symmetrical.
- We've done this before.
- This is actually a little redundant from other
- problems we've done.
- But if these are combined 5%, then they're saying that each
- of these are 2 and a half percent.
- Each of these are 2 and a half percent.
- So to answer their question, what is the probability that a
- randomly chosen fifth grade boy would be taller than 157.7
- centimeters, well, that's literally just the area under
- this right green part.
- Maybe I'll do it in a different color.
- This magenta part that I'm coloring right now, that's
- just that area, we just figured out it's 2.5%.
- So there's a 2 and a half percent chance we randomly find
- a fifth grade boy who's taller than 157.7 centimeters,
- assuming this is the mean, the standard deviation, and we are
- dealing with a normal distribution.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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