Transcript 5.1 day 2 - Amazon S3

5.1 day 2
Simulations! 
Review from 5.1
• To pass the time during a long drive, you and a
friend are keeping track of the makes and models
of cars that pass by in the other direction. At one
point, you realize that among the last 20 cars,
there hasn’t been a single Ford. (Currently, about
16% of cars sold in America are Fords). Your
friend says, “The law of averages says that the
next car is almost certain to be a Ford.” Explain to
your friend what he doesn’t understand about
probability.
Simulation
• An imitation of chance behavior, based on a model that
accurately reflects a situation.
• Statistics Problems Demand Consistency
• Recall the four step process:
• State: what is the question of interest about some chance
process?
• Plan: Describe how to use a chance device to imitate one
repetition of the process. Explain clearly how to identify the
outcomes of the chance process and what variable to measure
• Do: Perform many repetitions of the simulation
• Conclude: Use the results of your simulation to answer the
questions of interest.
Stoplight!!
• One her drive to work every day, Ilana passes through an
intersection with a traffic light. The light has probability 1/3 of
being green when she gets to the intersection. Explain how you
would use each chance device to simulate whether the light is red
or green on a given day.
a) A six-sided die
Let 1 and 2 represent a green light and 3-6 represent a red light. Roll the
die once.
b) Table D of random digits
Let the numbers 1, 2 and 3 represent a green light and 4-9 represent
a red light. Ignore 0. Look up one number in the table.
c) A standard deck of playing cards
Let diamonds represent a green light and clubs and spades represent
a red light. Deal one card. If a heart is dealt, ignore that outcome and
deal again.
Find the error!
• Explain what’s wrong with each of the
following simulation designs.
a) A roulette when has 38 colored slots – 18 red,
18 black, and 2 green. To simulate one spin of
the wheel, let numbers 00 to 18 represent red,
19 to 37 represent black, and 38 to 40 represent
green.
There are actually 19 numbers between 00 and 18, 19 numbers
between 19 and 37, and 3 numbers between 38 and 40. This
changes the proportions between the three different
outcomes.
Find the error!
• About 10% of US adults are left-handed. To
simulate randomly selecting one adult at a time
until you find a left-hander, use two digits. Let 01
to 10 represent being left-handed and 11 to 00
represent being right handed. Move across a row
in Table D, two digits at a time, skipping nay
numbers that have already appeared, until you
find a number between 01 and 10. Record the
number of people selected.
There is no reason to skip numbers that have already been encountered in the
table. These numbers just represent the handedness, not a particular individual
to select for the sample.
Is the simulation valid
17. Is this valid? Determine whether each of the
following simulation designs is valid. Justify your
answer.
a) According to a recent poll, 75% of American
adults regularly recycle. To simulate choosing a
random sample of 100 US adults and seeing how
many of them recycle, roll a 4 sided die 100 times.
A result of 1,2, or 3 means the person recycles; a 4
means that the person doesn’t recycle.
This is a legitimate simulation. The chance of rolling a 1, 2 or 3 is 75% on a
4-sided die and the rolls are independent of each other.
Continued
• b) An archer hits the center of the target with
60% of her shorts. To simulate having her
shoot 10 times, use a coin. Flip the coin once
for each of the 10 shots. If it lands heads up,
then she hits the center of the target. If the
coin lands tails up, she doesn’t.
This is not a valid design because the chance of heads is
50% (assuming the coin is fair) rather than the 60% that
she hits the center of the target. This will underestimate
her percent of hitting the target.
Airport Security
19. Airport Security. The Transportation Security Administration
(TSA) is responsible for airport safety. On some flights, TSA
officers randomly select passengers for an extra security check
prior to boarding. One such flight had 76 passengers – 12 in first
class and 64 in coach class. Some passengers were surprised
when none of the 10 passengers chosen for screening were seated
in the first class. We can use a simulation to see if this result is
likely to happen by chance.
a) State the question of interest using the language of probability.
What is the probability that, in a random selection of 10 passengers,
none from first class are chosen?
b) How would you use random digits to imitate one repetition of
the process? What variable would you measure?
Number the first class passengers as 01-12 and the other passengers as 13-76. Ignore all
other numbers. Look up two-digit numbers in Table D until you have 10 unique numbers (no
repetitions because you do not want to select the same person twice). Count the number of
two-digit numbers between 01 and 12.
• c) Use the line of random digits below to perform on
repetition. Mark directly on or above them to show
how you determined the outcomes of the chance
process.
71487 09984 29077 14863 61683 47052 62224 51025
The numbers read in pairs are: 71 48 70 99 84 29 07 71 48 63 61 68 34 70 52. The bold
numbers indicate people who have been selected. The other numbers are either too
large (over 76) or have already been selected. There is one person among the 10 selected
who is in first class in this sample.
• d) In 100 repetitions of the simulation, there were 15
times when none of the 10 passengers chosen was
seated in first class. What conclusion would you
draw?
Since in 15% of the samples no first class passenger was chosen, it seems
plausible that the actual selection was random.
Recycling!
• 23. Do most teens recycle? To find out ,
an AP statistics class asked an SRS of 100
students at their school whether they
regularly recycle. How many students in
the sample would need to say “Yes” to
provide convincing evidence that more
than half of the students at the school
recycle? The Fathom dotplot below
shows the results of taking 200 STSs of
100 students from a population in which
the true proportion who recycle is 0.50.
• a) Suppose 55 students in the class’s
sample say “Yes”. Explain why this result
does not give convincing evidence that
more than half of the school’s students
recycle.
43 of the 200 samples yielded at least 55% who say they
recycle. This means that, according to the simulation, if 50%
recycle, we would see at least 55% of the sample saying the
recycle in about 21.5%. This is not particularly unusual.
• b) Suppose 63 students in the class’s sample
say “Yes.”
Explain why this result gives strong evidence
that a majority of the school’s students
recycle.
However, only 1 of the 200 samples yielded at
least 63% who said that they recycle. This
means that it would happen in about 0.5% of
samples. This seems rather unusual. It would
be much more likely that the actual
percentage who recycle is larger than 50%.
Color Blind Men
About 7% of men in the US have some form of redgreen color blindness. Suppose we randomly select
one US adult male at a time until we find one who is
red-green color blind. How many men would we
expect to choose on average? Design and carry out
a simulation to answer this question. Follow the
four step process.
• State: How many men would be expect to choose in order to find one
who is red-green colorblind?
• Plan: We’ll use technology to simulate choosing men. We’ll label the
numbers 01-07 as colorblind men and all other two-digit numbers as
non-colorblind men. Use technology to produce two-digit numbers until
a number between 01 and 07 appear. Count how many two-digit
numbers there are in the sample.
• Do: We did 50 repetitions of the simulation using technology. The first
repetition is given here: 17 33 49 41 02. The number in bold is the
stopping point. For this repetition we chose 5 men in order to get the
one colorblind man. The dotplot below gives the number of men
chosen to get a colorblind man in each of 50 repetitions. The average of
the number of men from these samples is 16.88.
• Conclude: Based on our simulation, we would suggest that we would
need to sample about 17 men, on average. That is, we would have to
choose about 16 men before getting a colorblind man.
Common AP Error
• When making conclusions, students often lose
credit for suggesting that a claim is definitely
true or that the evidence proves that a claim is
incorrect.
• Instead, say that there is (or isn’t) sufficient
evidence to support the claim.
Assignment
• Page 295 15,17,19,23,25