Math 3307
Lecture Notes
Perkowsky text
Monday format
May’13
Jan. 2015
Chapter 7
Homework Assignments
10 points each problem
Homework 5
Chapter 7
70 points
2, 4, 6, 12, 14, 16, 20
Homework style sheet and rules:
Work on one side only; pdf it and upload it before the deadline on the calendar.
Work that is poorly scanned or illegible will be given a zero.
This includes sideways or upside down scans!
Do NOT crowd the work, leave at least 3” between problems.
Label the answers carefully so the grader can grade efficiently.
1
Chapter 7 – Random Variables and Probability Distributions
7.1
What is a Random Variable?
Technically:
A quantitative variable whose value is determined by the outcome of a chance
experiment.
The book’s example of a free throw is excellent! It starts on page 183…
Let’s discuss the Classroom Exploration on page 185 together
Why 6 blue/4 red…why not 5/5
And let’s check out the probability table! Would you have thought of a grid?
Which type of learner is likely to appreciate which presentation?
Can we turn it into a tree diagram?
NOTE TI simulation
page 185 This is VERY useful for making up
worksheets, quizzes, and tests.
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Now let’s turn the information into the most abstract representation of all:
Discrete Random Variable: a finite number or a countable number of outcomes.
Continuous Random Variable: inifinitely many variables, situated on a
numberline with no gaps or interruptions.
Which of the following are discrete? Continuous?
The number of eggs received by the shipping department at the local Krogers
on a given day.
The number of people marching in the Fourth of July parade downtown
Houston.
The measure of voltage for a smoke detector in your kitchen.
The temperature in Houston.
The exact playing time for a given baseball game.
The number of actors in a randomly selected movie.
The weight of a randomly selected human.
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Probability Distribution Table
page 186
Probability Density Table?
Review the rules, page 130!
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Problem 1
In a drug study, there is a control group and a group of people not taking the drug.
The drug is to help you have girls for children. These are VERY large groups.
Here is a table for the control group. Is it a Probability Distribution Table?
X
P(X)
0
.125
1
.375
2
.375
3
.125
What is X? Are all possibilities covered? How did they get those numbers? Do
they add up right?
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Let’s do the Focus on Understanding p.189 together. Read page 188 and we’ll
discuss the 3 questions in the box on page 189
With 2 dice and adding the up face, how will we create our sample space:
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Now let’s list outcomes and probabilities:
Now let’s look at that game! Is it fair? How can we make it fair if we need to?
Can we make an outcome grid?
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7.2
The Mean of a Random Variable
When we discuss the measure of center for a random variable, we’ll call it the
expected value (E(x)). It’s like a mean but in a different context. It is a long term
average value.
Let’s review the grid on page 186:
100 outcomes:
40
24
36
zero’s
one’s
two’s
If you’re Nicky’s coach you’ll have an expectation of how she’ll do in the free
throw situation…this expectation is the “mean”.
Of course there’s a formula!
(see the box, page 190)
Multiply each outcome value by the probability that it may happen.
Add.
0(40%)+1(24%)+2(36%) = 96%
this is the expected value for a trip to the line
Now let’s review:
What if Nicky were an 80% shooter?
Focus on Understanding
page 191
8
EV problem 1
Years ago, members of organized crime groups ran numbers games. Now such
games are legalized. New Jersey’s Pick 3 game works this way:
Bet 50 cents and select a three-digit number between 000 and 999. If your 3 digits
match the numbers drawn, you win $275.
Prob (winning):
Amount won:
Prob (losing):
Amount lost:
(remember to subtract the cost of the game from the winnings!)
Is this a fair game?
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EV Problem 2
The CAN Insurance Company charges Mike $250 for a one-year $100,000 life
insurance policy. Because Mike is a 21 year old male, there is a 0.9985 probability
that he’ll live for that year.
What are the outcomes and their probabilities?
What are the financial outcomes for each probability?
What is the expected value?
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The Prime Number Multiplication Game
page 191
Two number cubes! Multiply the top faces: do this and see what answers you get.
How many numbers will there be in the sample space? Ok group by group let’s get
the numbers identified and multiplied on this page:
On the next page we’ll identify them as prime or composite and calculate the probs.
Summarize in a frequency table:
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Which are the primes:
Which are the composites and “not primes”?
What is the probability of getting a prime?
One or composite?
Points for primes: 10
points for other: 1
Is it a fair game?
ACTIVITIES: How would you handle the points to make it as fair as possible?
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Hard choices!
Page 192
Let’s look at Tasa’s lawn mowing gig at the bottom of page 192. Analyze it in your
group.
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7.3
Variance and Standard Deviation
Measures of spread and variability – we have them in this context, too!
Let’s look at this from a vocabulary standpoint:
Deviation (from the mean):
(x - )
Squared deviation:
(x - )2
Let’s sketch this:
Note that the further from the mean a point is, the bigger the squared deviation!
Now let’s look again at Tasa’s possible earnings for mowing the grass again.
See page 193
See page 195 for the formula for variance (remember standard deviation is the
square root of variance!) Let’s decode it!
Calculate the squared deviation, multiply each times it’s probability…add them up.
See page 194, bottom, for Tasa’s variance.
See page 195 for an alternate version of the formula.
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For Option 2B, let’s check out the top of page 196 and do some comparisons.
Now let’s move into Standard Deviation.
What is standard deviation? Let’s go around the room and discuss what it is!
Now check out the computer instructions on the bottom of page 197. Please use
this in your homework and in class!
What is the z-score? Again, around the room…
Let’s draw some pictures of z-score
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Let’s check out using the TI for finding these numbers: page 198 – 199
We’ll do this with the lawn mowing data!
And for the sum of two cubes data from the 7.1 experiment.
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Let’s look at a sample with a mean of 5 and a standard deviation of 2…what can
we say about a measurement of 7 or 3 using the z-score of these?
ACTIVITIES
SD and ZS
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7.4
Binomial Random Variables
Often we have a situation with repeated identical trials. Tossing a free throw (it
goes in or it doesn’t), tossing a coin (heads/tails), landing a plane (ok/crash), having
a baby (boy/girl), taking a T/F test.
These trials need to be independent of one another! If they are, then we may
multiply the individual probabilities for the outcomes.
Let’s analyze the standard 2 child family:
The 3 child family:
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Note that we will be using COMBINATIONS when we count outcomes:
Let’s look at the 3 child family again:
BBB
GGG
2B1G
1B2G
The combination of 3 kids taken 2 at a time:
3
 3  3!
C 2   
3
 2  2!1!
P(3girls) = 3/8
Summary of Binomial Experiments/Probability: page 202 and 203
Let’s review it carefully!
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The complement rule:
Application of it: page 204 middle, gray box
Are there mean, variance, and standard deviation? Bet your grade on it.
Summary page 205, bottom, box
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TI – let’s learn how to do this efficiently: page 206
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Now we know it’s binomial: check those possibilities again with the formula.
EXAMPLE
In a drug study, there is a control group and a group of people not taking the drug.
The drug is to help you have girls for children. These are VERY large groups.
Here is a table for the control group. Is it a Probability Distribution Table?
X
P(X)
0
.125
1
.375
2
.375
3
.125
What is the Expected Value? Mean? Standard Deviation?
What does the histogram look like?
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Which of the following are binomial experiments?
Surveying 1000 people and asking them to rate the president on a scale of 1 – 5
Rolling a fair die 50 times
Having kids
Determining whether 12,000 pacemakers are defective or not, one by one
Guessing on a T/F test
Guessing on a test with 5 answer choices per question
ACTIVITIES BIN PROB using your calculator
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BP Problem 1
Bob is a self-proclaimed mentalist who claims he can read minds. To test this, he is
given 14 T/F questions.
A.
He gets 8 of them right. What is the expected value and is this unusual?
B.
He gets 11 of them right. What is the expected value and is this unusual?
C.
He gets 2 of the right? EV is? Is this unusual?
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BP Problem 2
There is a 0.723 probability that an airplane will land on time at Hobby. Discuss
whether that result would be considered unusual or normal.
A
Find the probability that at least 5 out of 6 airplanes arrive on time in a given
period of time.
B
Find the probability that at most 2 airplanes arrive on time in a given period
of time.
C
Find that probability that exactly 3 airplanes land on time in a given period of
time.
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BP Problem 3
Internal surveys show that directory assistance providers give the wrong number
15% of the time. Assume you are testing a provider by making 10 requests.
Assume further that this is a very average company and gives wrong answers 15%
of the time.
Find the probability of getting one wrong answer. Is this unusual?
Find the probability of getting at most one wrong answer. Is this unusual?
Is the probability really 15% for this company?
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BP Problem 4
A study was conducted to determine whether there were significant differences
between medical students admitted through special programs and medical students
admitted through the regular admissions criteria. It is claimed that the graduation
rate for the students admitted through the special programs is 94%.
If 10 students from the special programs are randomly selected, find the probability
that at least 9 of them graduated.
Would it be unusual to randomly select 10 and find that 7 graduated? Why or why
not?
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7.5
The Normal Curve
The standard normal curve
the bell curve
symmetric, mound shaped,
continuous
Let’s discuss continuous versus discrete
For the standard normal curve the mean is zero and
the standard deviation is 1.
It is symmetric about z = 0…not x? why not?
Probabilities correspond to area under the curve.
Let’s review the Empirical Rule (p. 71) right now with a picture:
ACTIVITIES - ER
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Now let’s look at the standard normal probability table.
Given a z-score of 1.28, what is the probability that a measurement is at or below
this value?
page 210
Now for using the chart with “greater than or equal to”…a version of the
complement rule!
Or between two measurements!
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Using the table in reverse: from a probability to a z-score:
Page 212
In reality, MOST normal curves are NOT standard! How do we rescale to make
use of our standard normal chart? With z-scores! All normal curves are
proportional and we use the z-score calculation to make them “fit” the table.
Page 213
Let’s look at a normal curve with mean 5 and sd .75. What’s the area between 1 sd
above and below the mean? Empirical Rule. What are those measurements for
THIS curve. How will we use z-score to discover this on the chart for the standard
normal distribution?
ACTIVITIES – STD Normal Table
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Focus on Understanding:
page 215
Using the TI to do this, chart-free! Pages 216 – 218
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From another source – TI83 instructions for
Areas between two bounds:
2nd VARS
[2: normal cdf(left z score, right z score)]
Normal Distributions:
The Precision Scientific Instrument Company manufactures thermometers. To
check the accuracy, they test the thermometers in freezing water and make sure it
registers 0 degrees F. Of course some are high and some are low. Assume there is
a standard deviation of 1 degree F. Find the area and show it on a standard normal
curve!
What is the probability that the reading is less than 1.58°?
You should get 94.29%
What is the probability that the reading is above −1.23°?
You should get .8907
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What is the probability that the reading is between −2° and 1.5°?
You should get 91.04%
Working backwards in the chart:
Find the temperature associated with the 95th percentile. z = 1.645
How does this work?
Find the temperatures separating the bottom 2.5% and the top 2.5%
These are called tolerances. (−1.96 and 1.96 for z’s).
How does this work?
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Fill in the blanks:
About _________% of the area is within 1 standard deviation of the mean
About _________% of the area is within 2 standard deviations of the mean
About _________% of the area is within 3 standard deviations of the mean
ACTIVITIES
Find the probabilities:
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Find the following percentiles:
P95
P75
P50
P35
Enrichment:
c
5
10
Here is a probability distribution. Find the value of c.
Find the probability that x is between 0 and 3.
Find the probability that x is between 2 and 9.
ACTIVITIES – review, UNIFORM Distribution
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ND problem 1
Air Force ejection seats are designed for people weighing between 140 lb and 211
lb. Women’s weights are normally distributed with a mean of 143 lb and a standard
deviation of 29 lb. What percentage of women have weights in those limits?
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ND problem 2
ACTIVITIES
ND problem 3
The lengths of pregnancies are normally distributed with a mean of 268 days and a
standard deviation of 15 days.
A woman wrote to Dear Abby claiming that she gave birth 308 days after a brief
visit with her husband who was fleet Navy and ship bound else. Is this credible?
Premature is being born in the 4th percentile of length of pregnancy…what length of
time is this?
Can you figure out how we could use this fact to help hospital administrators?
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7.6
Normal Approximations
Sometimes, when everything is right, you may approximate a binomial
distribution as a normal distribution and use the far easier calculations for the
normal distribution.
What is “everything is right”?
1.
the binomial distribution is “smooth” , not “chunky”
2.
the binomial distribution is symmetric, not skewed
3.
The number of data points times the minimum (p, q) > 5
If you’ve got these three things you are good to approximate.
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Now, we can look at the TI way to do this (page 221, bottom). Let’s compare with
another way on page 222. We’ll use the “continuity correction” (page 223) with
abandon! It’s a sort of “split the difference” way to manage the discrete nature of
real binomial data!
Let’s go through the calculations on pages 224 – 226
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Now let’s look at “tossing tacks” page 227
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Approximating normal
When an airliner is loaded with passengers, baggage, and cargo plus fuel, the pilot
must verify that the gross weight is below a maximum and that the weight is
properly distributed for safety.
An airline has established a procedure in which extra cargo must be eliminated
whenever a 200 person plane has at least 120 men.
Assume that the population is 50/50 men and women.
Check to make sure we can approximate:
Get the mean and standard deviation:
Mean:
Sigma:
did you get 7.0710678?
Continuity Correction:
119.5 to 120.5
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We want “at least 120 men”…120 and to the RIGHT…sketch this!
Now find the area that is shaded.
z = 2.76
What is the probability? Do we need to worry much about this?
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Using continuity corrections:
Wording:
At least 120
to the right of 119.5
More than 120
to the right of 120.5
At most 120
to the left of 120.5
Fewer than 120
to the left of 119.5
Exactly 120
between 119.5 and 120.5
Let’s look again at page 223 and the Continuity Correction. And at the properties
of Binomial Distributions that can be approximated as Normal (page 219)
Alternate ACTIVITIES and the following problems! Talk out loud! And Share!
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AN Problem 1
In a study of 420,000 cell phone users in Denmark, it was found that 135 developed
brain cancer. Assuming cell phones have no effect, there is a 0.000340 probability
of a person developing brain cancer. We would, then, expect 143 cases among
420,000 randomly chosen people.
Estimate the probability of 135 or fewer cases of such cancer in the randomly
chosen population.
What do these results suggest about media reports that cell phones cause brain
cancer?
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AN Problem 2
After being rejected for employment, Ms. Kim learns that this company has hired
only 21 women applicants among its 62 new employees. She also learns that the
pool of applicants is very large with equal numbers of qualified men and women.
The company claims no unfair discrimination in hiring. Kim feels differently.
Run the numbers and decide how you feel.
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AN Problem 3
45% of humans have Type O blood. A hospital is running low on Type O blood
and runs a blood drive…it needs 177 units of this type of blood. Assume 1 unit per
donor. If 400 volunteers show up, what is the probability that at least 177 of them
will have Type O blood? Are the 400 volunteers enough?
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