Transcript Discrete variables

AP Statistics B
March 1, 2012
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AP Statistics B warm-ups
Thursday, March 1, 2012
You take your car to the mechanic for a steering
problem (welcome to adulthood, BTW, and
don’t forget to pay for your car insurance). The
mechanic says he may be able to fix the problem
by putting the car on the jack and whacking it
with a hammer, in which case he’ll charge you
$50. That is successful about 40% of the time. If
it’s not successful, he’ll have to spend $200 for
parts and $300 for labor. Use the expected value
model to determine the average (probable)
costs a driver will incur in this situation.
(Solution on the next slide)
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Answers to AP Statistics B warm-ups
Thursday, March 1, 2012
1. Determine the probability model: 40%
chance of a $50 repair, or what percentage of
the other repair? It has to be 60% (the
remainder percentage necessary to add up to
100%) and the $500 in costs ($200 in parts
and $300 in labor)
2. Calculate the expected value: .4($50) +
.6($500) = $20 + $300 = $320.
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Comments about the meaning of
“expected value” in this problem
(audio only)
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Outline for materials in Chapter 16
1. Vocabulary:
1. Discrete variable
2. Continuous variable
3. Probability model
2. Expected value
3. Using the TI 83+ to calculate expected values
4. Application of the probability model and
calculation of variance
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Discrete v. continuous
• Discrete variables—basic idea is that discrete
variables can be counted (i.e., are not infinite)
• Continuous variables—infinitely many, cannot
be counted
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Examples of discrete variables
• Height and weight charts for a given
population
• Ages
• SAT scores (example of when discrete math is
restricted to whole number or integer values
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Examples of continuous variables
• Take, for example, a simple line: y=x
• Assume we’re measuring distance travelled,
and that the object is travelling at 1 ft/sec
• Let’s look (on the next slide) at what the graph
of that would look like.
• This will be an example of a continuous
variable.
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Graph of a continuous variable
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Comparing discrete and continuous
Most statistical data is discrete, not continuous.
Raw data, even in a normal curve, looks like
this:
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We can approximate the normal
histogram by a continuous curve
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The normal equation can be
algebraically manipulated, but it
is….well, see for yourself:
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P( X ) 
 2
 x
e
( )
2
2
2
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However, it does produce the z-tables
and a graph that we can analyze easily
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Probability model
• Fancy term, simple idea
• “Model” means a theory that predicts
outcomes (used in engineering, science, all
social science, business, etc.)
• Per the text (p. 369), a probability is “the
collection of all the possible values and the
probabilities that they occur”
• Let’s apply this definition to a couple of
models that we’ve already seen
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Probability model: the lottery example
• Remember the lottery we did a couple of days
ago (I’m picking the first one):
– 14 $5-bills
– 2 $10-bills
– 4 $20-bills
• The values (i.e., what bill you could draw out of
the bag) is $5, $10, and $20. There are no other
possible outcomes under the rules of the game
• Their probabilities are 14/20, 2/20, and 4/20,
respectively (aka 0.70, 0.10, and 0.20)
• Together, these 6 data form the probability model
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Probability model: the lifeinsurance/disability model on pp. 369-70
• The possible outcomes are death, disability or
nothing
• The values associated with each outcome are
$10,000, $5,000, and $0.
• The probabilities associated with each
outcome are, respectively, 1/1,000, 2/1,000,
and 997/1,000
• Put together we can establish the average
value of the policy or the cost to the company.
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Expected value
• We reviewed the expected value of a probability
model yesterday: E(X)=∑xi∙P(xi)
• What I forgot to mention was something very
important, namely that E(X)=μ.
• And μ is what? It is the mean of the population
(pronounced “mu” or “myu“, depending on
whether you like to make a bovine sound.
• This is an extremely important relationship, so
let’s explore its implications.
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Identity of expected value and mean
• Let’s explore this with the first lottery example
we did. Get out your calculators.
• In one of the lists, enter all the possible outcomes
of the lottery example we just did (i.e., enter 14
5’s, 2 10’s, and 4 20’s)
• Alternatively, if you enjoy busy work, add the
following and divide by 20:
5+5+5+5+5+5+5+5+5+5+5+5+5+5+10+10+20+20+
20+20
• If you did a list, use the 1-var method to find the
mean.
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Using the TI Tips on p. 372
• You will need to enter two lists. The first is 5,
10, 20. I’m going to call it L1, but you can use
any name you like as long as you remember it
• The second list, which I’ll call L2 (same caveat
as above), consists of the following 14/20,
2/20, 4/20 (note that the List functions
accepts fractions and calculates decimals)
• The next slide will show you how to calculate
the expected value (which equals what?)
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Using VarStats
• Here, the book is genuinely confusing, though it
doesn’t mean to be
• It looks like it wants you to run VarStats while
subtracting it from 1: “ask for 1-VarStats L1, L2”
• What they really want is different:
–
–
–
–
Press the STAT button.
Select the CALC menu from along the top
Select the first entry, which is listed as “1-Var Stats”
Put “L1, L2” (or whatever you used) after it and hit the
enter key
• How does this answer compare to the mean you
calculated by hand? (should be quite similar….duh)
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Calculating variance and standard
deviation by hand under the expected
value model
• Read pp. 370-71 “First Center, Now Spread….”
• When you’re done, move on to the next slide.
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Tedious, but necessary
• You’re going to have to be able to apply this
formula on some of the problems:
Var(X)=σ2=∑(x-μ)2∙P(X)
• Yes, you WILL have to subtract the mean from
the same entries. Así es la vida…large y dura.
• Fortunately, calculating the standard deviation
(σ) is simply a matter of taking the square root
of the variance (Var(X))
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Meaning of standard deviation in this
context
• What does standard deviation mean in for the
normal distribution?
• Here, it means something different: a way of
evaluating how wild are the variations.
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Doing the problem on pp. 371-72
• Somebody read the problem out loud.
• Work through the problem.
• I’ll give you my analysis on the next slide, but
keep the tree diagram on p. 371 in mind.
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Analyzing the problem
• Confusing nomenclature without explanation:
– NN means the client got two new computers
– RR means they got 2 refurbished computers
– RN and NR are the probabilities that they got one
new and one refurbished computer
• Breaking the tree diagram down
• Calculating variance on p. 372
• Explanation “in context”
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Homework, due Friday, March 2, 2012
• Chapter 16, problems 8, 11, 14, 17, 20, 23
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