Chapter 17 Answers to questions 15, 16, 17, 18 and 20

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Chapter 17
Answers to questions 15, 16, 17, 18
and 20
AP Statistics B
Overview
• This is the introduction to using the binomial
theorem
• I will do some of the calculations by hand
• Others, we will use the binomial function on your
calculator (“about $#%^@! time, “I hear you say,
though it was in the text which you could have
read…..)
• I will show you how the calculator works by doing
some examples that we first will work through by
hand
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Chapter 17, Exercise 15
• This is the first time we’ve been using the
binomial theorem to calculate mean and
standard deviation
• Set-ups and 15 and 16 are straightforward
• Remember Exercise 13 on lefties (lefthandedness, not political orientation):
p=0.13, q=0.87, n=5
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Chapter 17, Exercise 15(a) and (b)
• “How many lefties do you expect?” in (a) asks
for the mean. (b) is expressly the standard
deviations.
• Formulas:
Mean:
SD:
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Chapter 17, Exercise 15(c)
• “If we keep picking people until we find a lefty,
how long do you expect it will take?”
• TERRIBLE wording of the question: where
have we been dealing with TIME? (“how
long….”)
• What they REALLY want is the expected value,
which is what for the binomial distribution?
(next slide has answer, but quiz yourselves
first and come to a consensus)
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Chapter 17, Exercise 15(c)
(continued)
Answer:
μ = 1/p = 1/.13 = 7.69 people
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Chapter 17, Exercise 16
Virtually the identical set-up as Exercise 15. I am
therefore not going to repeat the workout, but
just give you the answers.
a. μ= 4.8
b. σ=0.98
c. 1.25 shots
Confused? Email me!
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Chapter 17, Exercise 17
• Same set-up as 15, except we increase n from 5
to 12
• p still is 0.13, q is 0.87, n=12.
• Is this correct?
• Well, not exactly….because the questions asks for
mu and sigma (mean and standard deviation) for
the number of RIGHT-HANDERS in the group
• So p becomes q and q become p, since success is
finding RIGHT-HANDERS
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Chapter 17, Exercise 17(a)
Right-handed mean and standard deviation
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Chapter 17, Exercise 17(b)
• What’s the probability that they’re not all
right-handed?
• Trick: find the percentage of all right-handers,
and subtract from 1.
• First calculate:
• Then subtract from 1: 1-0.188=0.812
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Chapter 17, Exercise 17(c)
• “What’s the probability that there are no more than 10
righties?”
• Think before calculating.
• Do you really want to calculate all the following binomial
coefficients? (The answer is “no”, in case you’re wondering.)
Discuss among yourselves alternatives to such massive
calculations, and when you reach a consensus, go on to the
next slide.
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Chapter 17, Exercise 17(c)
Alternatives to brute-force calculation
• We do what we did before: find the
percentage MORE than 10, and calculate that
• It’s considerably easier, since we only have
two binomial coefficients to calculate:
• The second is just 1, so let’s calculate the first:
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Chapter 17, Exercise 17(c)
Putting everything together
• We have the binomial coefficients, so all we
have to do is connect them to the terms they
belong with.
• Don’t forget to subtract this from 1! (Why?)
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Chapter 17, Exercise 17(d)
• The probability of exactly 6 of each?
• So n=12, k=16, and we calculate:
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Chapter 17, Exercise 17(e):
Time to throw in the towel on computation
• OK, I surrender to technology
• Yes, we COULD calculate the answer to this. But we’d
have to calculate each of the following binomial
coefficients (or generate Pascal’s triangle for n=12),
and pair them with the appropriate exponentiated
variables, then add them up:
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Chapter 17: calculating binomials with
the TI 83+/84+ calculator
• As Monsanto used to say, “better living
through chemistry”
• We’re doing better living through calculators
(for once)
• You need to learn about the binomial
distribution
• On the next page are pictures of your
calculators showing you what button to push
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Calculating binomials with TI 83+/84+
Using the 2ND-DISTR key combination
Location of keys on TI 83+
Location of keys on TI 84+
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Getting to the binomial distributions
Keys to press
• Push the 2nd-Distr keys
together (Distr is on the
VARS key, just below the
cursor keys)
• You should get a screen that
looks like this.
• Move to the binom
distributions by pushing the
top cursor key to get this
screen
Screen of the TI
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binompdf and binomcdf
• Here are the two functions
we’ll be working with
• binompdf is “binomial
probability density function”
which we use to determine
probabilities of a single
parameter (e.g., 5 out of 12)
• binomcdf=“binomial
cumulative density function”,
used to find things like “find 4
or fewer hits out of 12”
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How to use binompdf
• Let’s redo Exercise 17(d)
• n=12, p=0.87, and x=6 (i.e., a population of 12,
p of 0.87, and 6 out of 12 are right-handed
• Answer: 0.00193
• On the next slide, I’ll take you through how to
do it on the calculator.
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binompdf
n=6, p=0.87, x=6
• Choosing binompdf
gives you a screen like
this
• Fill it in like this:
n
p
x
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binompdf
• Press ENTER and….
• Pretty easy, huh?
• MUCH easier than
doing it by hand.
• This is how we’ll do the
rest of them
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Recap on binompdf
• Use to solve questions like “exactly 6 of 13
have are left-handed”
• Access through VARS key, but be sure you
press 2ND-VARS (actually 2ND-DISTR)
• Syntax is binompdf(n, p, x)
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Introduction to binomcdf
• binomcdf=binomial cumulative density function
• Think of this as being like what we did with the normal
function, except easier
• Remember how we’d say on the normal model that the
percentage of the population below Z was x%?
• It will be similar here: this function answers questions
like “what’s the probability of finding 5 left-handers or
less in a population of 9?”
• Good news: we don’t have to calculate all the binary
coefficients and hook them up with the probability
products of success and failure
• Bad news: well, see the next slide
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binomcdf: good news and bad
• Good news: easy to calculate IF set up right
• Bad news: not always set up right.
• For example, Exercise 17(e): “What is the
probability that the MAJORITY is right handed?”
• This is the equivalent of asking for 1-(Probability
of 6 or fewer right-handers)
• Good news (redux): we can set up the equation
like this: answer=1-binomcdf(12, 0.87, 6)
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binomcdf: Exercise 17(e)
• Choose binomcdf
• Enter values for n, p, and x and
press enter
• Not so hard, is it? Except that
we forgot to subtract this from
1!
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binomcdf: subtracting
• Here’s the way you
SHOULD set it up
• Set up, as usual, is
harder and more
challenging than the
calculation…..
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Redoing Exercises 17(b) and (c)
using the TI
• Look back on 17(b) and (c)
• See if you can write out ways of calculating
those answers by using either binompdf or
binomcdf
• Take 5-6 minutes to practice.
• Share at your table, then share out to the class
• Somebody write the collective wisdom on the
board, then go on to the next couple of slides
for the answers
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Redoing Exercises 17(b) and (c)
using the TI (answer to set-up)
b. probability that not all are right-handers =
1-binompdf(12, .87, 12)
OR binomcdf(12, .87, 11)
c. probability that no more than 10 righties =
binomcdf(12, .87, 10)
d. we’ve already redone (d), right?
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Answers
b. 1-binompdf(12, .87, 12)
or binomcdf(12, .87, 11)
c. probability that no more
than 10 righties =
binomcdf(12, .87, 10)
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Using the TI for Exercise 18
• Enough of working by hand. Technology is a tool. You
shouldn’t try to hammer a nail in with your hand or
your neighbor’s head.
• Data for Exercise 18 on our archer Diana:
n=10
p=.80 (would Jonathan Zuniga or Jose
Fuentes be this successful? Don’t believe
them….)
q=.20
Do the problems together using your calculators. Answers
with screen shots are on next slides
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Answers, Exercise 18(a)-(c)
a. μ= 8
σ=1.26
b. Probability that never
misses=10 of 10
hits=use binompdf
c. No more than 8 bull’s
eyes: use binomcdf
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Answers, Exercise 18(d)-(e)
d. Exactly 8 bull’s
eyes=use binompdf
e. Hits bull’s-eye more
often than she misses:
use binomcdf, but
subtract from 1:
1-binomcdf (5
successes)
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Exercise 20
Set-up on the TI
• Success=presence of the trait; failure=lack of trait (or
vice versa, as we’ve seen)
• Success=1 out of 8. What’s that in percentages? 12.5%
or 0.125 for success, 0.875 for failure
• n= 12
• (a), (b), and (d) are fairly straightforward, though you
can do (a) a couple of different ways.
• (c) is something we haven’t done yet, using the TI,
anyway, so you should stop at that slide and watch
carefully.
• Except for(c), I’m just going to show you what to use
and how to set it up, and give you screens shots of the
answers
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Exercise 20(a)
Two options, for once
(a) No frogs with traits…
use either binomcdf or
binompdf
Why can you use either to
get the same answer?
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Exercise 20(b)
Subtraction and reframing again…
• “at least 2 frogs” means we
could have 2, 3, 4, 5,6,7, 8,
9, 10, 11 or 12 frogs with
the unusual trait
• We already have the figure
from 20(a) for 0 frogs with
condition (=.201). Add that
to the probability of 1 frog
with condition, and you
have the probability of
exactly 1 frog having the
condition
• Subtract this sum from 1 to
get your answer
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Exercise 20(d)
Straightforward “plug and chug”
(d): “no more than 4 frogs
with the (1 out of 8) trait”
Obvious and straightforward
application of binomcdf
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Finally, 20(c)
Sum of two probabilities
• “3 or 4 frogs” with the
trait
• Easiest approach here is
to use binompdf twice:
one for exactly three
frogs and one for
exactly 4 frogs
• Answer: binompdf(3
frogs with
trait)+binompdf(4 frogs
with trait)
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