Chapter 17: The Binomial Model.
Part 4
When to substitute the normal
model
AP Statistics B
Outline of today’s class
• Problems/limitations with the binomial model
• Alternatives to the binomial model that are
easier to calculate
• Applications of the alternatives when we have
large populations (large n, that is)
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Warmup, Statistics
1. Write out, but DO NOT CALCULATE getting
exactly 247 heads on 500 flips of an unbiased
coin using the binomial model.
Answer these among yourselves and agree on
an answer. Worked out problem are on the
next slides.
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Answer to warm-up
• What’s the formula we use?
• So for this case, the answer would be
• Too much to calculate! (If you tried to calculate,
please report immediately to the nurse to be
examined…..)
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The binomial model:
Limitations
• Great for small numbers (less than 20)
• Really REALLY bad for big numbers (500! has
about 100,000 digits to it)
• Complicated calculations
• So, what’s a brother to do?
• What LOOKS like a binomial distribution?
• Well, let’s graph a few and see what we find….
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Binomial model: graphs (1)
• The graph on the right
is a typical example,
here where n=21
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Binomial model: graphs (2)
• Here’s a more typical
example, one close to
what we’ve worked
with (n=15)
• Here, p=1/3, which
means what?
• That we’re successful
one-third of the time,
or one time out of
three.
• Note that the curve is
skewed—which way?
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What do these curves look like?
•
•
•
•
•
•
Chapter 6, maybe?
Et seq.?
They have means and standard deviations?
Yes, it’s our old friend (drum roll please……)
THE NORMAL DISTRIBUTION!
Let’s compare the two……
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The binomial model:
Comparing binomial and normal distributions
Binomial (discrete)
Normal (continuous)
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OK, so they’re alike. How do we get
from one to the other?
• Well, what can you calculate using the
binomial formula and what you know about
the binomial distribution?
• Mean?
• Standard deviation?
• So take a minute, review your notes, and have
somebody write the mean and standard
deviation for a binomial distribution!
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The binomial model:
• That’s right; here they are:
– Mean:
– Standard deviation:
• These are REALLY important for you to know
cold (i.e., forget some of the lyrics to your
favorite song from 7th grade and remember
these guys instead)
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The binomial model:
converting and to normal values
• Compute and using the formulas just listed
– For this case we have μ=n ×p=500 ×½=250
– σ is a bit more complex:
• OK, so now what??????
• Now we have a mean and a standard
deviation and a normal model:
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The binomial/normal model:
what’s next?
• Remember the formula?
• Look back in your notes, and somebody write
it on the board.
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The binomial/normal model:
z-formula
• That’s right:
• For N(250, 11.18), what does this look like?
– Y is 247
– Mu is 250
– Sigma is 11.18. Except what’s the problem with this
analysis? Discuss among yourselves and come to a
conclusion
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The binomial/normal model:
z-formula
• The problem is that this measures the areas to
the LEFT of y, right?
• So we don’t get an exact calculation.
• How can we approximate?
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The binomial/normal model:
z-formula
• We approximate by calculating an interval.
• Example: use y=246.5 for one calculation, and y=
247.5 for the second, and subtract the first from
the second
• Not precise, but very VERY close.
• Most problems are easier. Let’s look at the one in
the book.
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Practice (Red Cross example):
The Binomial Horror
P. 394 of the text: Red Cross looking for 0-negative donors.
p=0.06 and n=32,000. Red Cross needs at least 1850 pints
(each person donates 1 pint; any more and you get really
dizzy, not to say ill)
Using the binary distribution, you have to calculate each y from
1850 through 32,000. That is, you’re going to have 32,0001850+1 calculations=30,151 calculations that look like this:
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Practice (Red Cross example):
Normal curve to the rescue
• Here, the normal curve really saves us
• Mean is np, or 32,000×0.06=1920
• Standard deviation is
• So now we can apply our z model and get
• Our tables or calculators tell us that only about
0.05 lie below this point, and that’s your answer.
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Practice:
How to apply this procedure
• If n>9, you can use the normal model.
• Calculate
• Adopt normal model as
• Apply normal formula (y-μ)/σ and solve as usual.
• Warning: be sure you know whether you’re
looking above or below y
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Practice:
Chapter 17, problem 22 (p. 399)
• Have someone read the problem
• Work together at your tables to solve the
problems. Everybody needs to be involved; no
free riders (pace Hickman, although in my
experience most libertarians don’t believe in free
riders whereas lawyers always do)
• Share answers among yourselves and reach a
consensus
• When you’re ready, move on to next slide for my
analysis
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Practice:
Problem 17(a)
•
•
•
•
This is a standard plug-and-chug problem
We know that
Here, n=200, p=0.8, and q= 0.2
So μ=160 and σ=5.66
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Practice:
Problem 17(b)
• No problem with using the normal model
here. Why? (you can find the answer in the
text on p. 394. If nobody knows, find it and
read it now!)
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Practice:
Problem 17(c)
• This is a common question you should expect on any
statistics exam. It’s also easy to answer.
• Take the mean, and add and subtract one standard
deviation. Repeat for two- and three-standard
deviations.
• In notation, calculate μ±σ, μ±2σ, and μ±3σ
• These correspond to the 68-95-99.7 model
• Numerical answers:
– 68% of the time she gets between 154.34 and 165.66 bull’s
eyes
– 95% of the time between 148.68 and 171.32 bull’s eyes,
and
– 99.7% of the time between 143.02 and 176.98 bull’s eyes
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Practice:
Problem 17(d)
• Use the answers from (c) to answer this
question.
• We know that 99.7% of the time, the archer
will get between 143.02 and 176.98 bull’s eyes
• If she gets 140 bull’s eyes or less, that’s only
0.0015%......VERY rare.
• One more slide for today
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Homework:
• Chapter 17, Exercises 24, 26, 27, 34 and 36.
• Read Chapters 18 and 19….we will be going
REALLY fast from now on.
• (Yes, I know: “eez now time for fiendish plan!”
which is from Boris Badenov of “Rocky and
Bullwinkle”, which I expect you to become
familiar with)
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