Download Edwards

Math 301
Introduction to Probability and Statistics
Spring 2013
Section 001 9:10 to 10:10, M W F
Instructor: Dr. Chris Edwards
Phone: 424-1358 or 948-3969
Office: Swart 123
Classroom: Swart 14 Text: Probability and Statistics, 8th edition, by Devore. Earlier editions
of the text should be acceptable.
Required Calculator: TI-83, TI-83 Plus, or TI-84 Plus, by Texas Instruments. Other TI
graphics calculators (like the TI-86) do not have the same statistics routines we will be using and
may cause you troubles.
Catalog Description: Elementary probability models, discrete and continuous random
variables, sampling and sampling distributions, estimation, and hypothesis testing. Prerequisite:
Mathematics 172 with a grade of C or better.
Course Objectives: The goal of statistics is to gain understanding from data. This course
focuses on critical thinking and active learning. Students will be engaged in statistical problem
solving and will develop intuition concerning data analysis, including the use of appropriate
technology. Specifically students will develop
•
an awareness of the nature and value of statistics
•
a sound, critical approach to interpreting statistics, including possible misuses
•
facility with statistical calculations and evaluations, using appropriate technology
•
effective written and oral communication skills
Grading: Final grades are based on these 300 points:
Exam 1
Topic
Summaries, Probability
Points
56 pts.
Tentative Date
March 4
Exam 2
Exam 3
Group Presentations
Homework
Distributions
Inference
15 Points Each
8 Points Each
56 pts.
56 pts.
60 pts.
72 pts.
April 12
May 10
Various
Mostly Weekly
Chapters
1, 2, 3.1 to 3.3,
4.1 to 4.2
3, 4, 5
7, 8
Grades: Grades will be assigned by the following schedule.
Grade
A
AB+
B
Points (Percent)
270 (90 %)
261 (87 %)
249 (83 %)
240 (80 %)
Grade
BC+
C
C-
Points (Percent)
231 (77 %)
219 (73 %)
210 (70 %)
201 (67 %)
Grade
D+
D
DF
Points (Percent)
189 (63 %)
180 (60 %)
171 (57 %)
170 or fewer
Homework: I will collect three homework problems approximately once a week. The due dates
are listed on the course outline below. While I will only be grading three problems, I presume
that you will be working on many more than just the three I assign. I suggest that you work
together in small groups on the homework for this class. What I expect is a well thought-out,
complete discussion of the problem. Please don’t just put down a numerical answer; I want to
see how you did the problem. (You won’t get full credit for just numerical answers.) The
method you use is much more important to me than the final answer. Important Grading
Feature: If your homework percentage is lower than your exam percentage, I will replace your
homework percentage with your exam percentage. Therefore, your homework grade cannot be
lower than your exam grade.
Presentations: There will be four presentations, each worth 15 points. The descriptions of the
presentations are in the Day By Day Notes. I will assign you to your groups for these
presentations as I want to avoid you having the same members each time. I expect each person
in a group to contribute to the work; however, you can allocate the work in any way you like. If
a group member is not contributing, see me as soon as possible so I can make a decision about
what to do. The topics are: 1 – Data Displays (February 15). 2 – Probability (March 1). 3 –
Central Limit Theorem (April 19). 4 – Statistical Hypothesis Testing (May 8).
Office Hours: Office hours are times when I will be in my office to help you. There are many
other times when I am in my office. If I am in and not busy, I will be happy to help. My office
hours for Spring 2013 semester are 10:20 to 11:00, Monday, Wednesday, and Friday or by
appointment.
Philosophy: I strongly believe that you, the student, are the only person who can make yourself
learn. Therefore, whenever it is appropriate, I expect you to discover the mathematics we will be
exploring. I do not feel that lecturing to you will teach you how to do mathematics. I hope to be
your guide while we learn some mathematics, but you will need to do the learning. I expect each
of you to come to class prepared to digest the day’s material. That means you will benefit most
by having read each section of the text and the Day By Day notes before class.
My idea of education is that one learns by doing. I believe that you must be engaged in the
learning process to learn well. Therefore, I view my job as a teacher not as telling you the
answers to the problems we will encounter, but rather pointing you in a direction that will allow
you to see the solutions yourselves. To accomplish that goal, I will find different interactive
activities for us to work on. Your job is to use me, your text, your friends, and any other
resources to become adept at the material.
Monday
Wednesday
Friday
January 28 Day 1
Introduction
January 30 Day 2
Random Sampling
Section 1.1
February 1 Day 3
Graphical Summaries
Section 1.2
February 4 Day 4
Graphical Summaries
Section 1.2
February 6 Day 5
Numerical Summaries
Sections 1.3 to 1.4
February 8 Day 6
Homework 1 Due
Intro to Probability
Sections 2.1 to 2.2
February 11 Day 7
Permutations, Combinations
Section 2.3
February 13 Day 8
Permutations, Combinations
Section 2.3
February 15 Day 9
Presentation 1
February 18 Day 10
Trees
Section 2.4
February 20 Day 11
Homework 2 Due
Bayes’, Independence
Section 2.5
February 22 Day 12
Coins, Dice, and RV’s
Sections 3.1 to 3.2
February 25 Day 13
RV’s and Expected Values
Section 3.3
February 27 Day 14
Homework 3 Due
Continuous Distributions
Sections 4.1 to 4.2
March 1 Day 15
Presentation 2
March 4 Day 16
Normal
Section 4.3
March 6 Day 17
March 8 Day 18
Normal Problems
Section 4.3
March 11 Day 19
Gamma
Section 4.4
March 13 Day 20
Probability Plots
Section 4.6
March 15 Day 21
Homework 4 Due
Binomial
Section 3.4
March 25 Day 22
Binomial
Section 3.4
March 27 Day 23
Hypergeometric
Section 3.5
March 29 Day 24
Homework 5 Due
Negative Binomial
Section 3.5
April 1 Day 25
Normal Approx to Binomial
Section 4.3
April 3 Day 26
Cauchy
April 5 Day 27
Linear Combinations, Central
Limit Theorem
Section 5.3
Exam 1
April 8 Day 28
Homework 6 Due
More CLT
Sections 5.4 to 5.5
April 10 Day 29
Catch up Day
Chapter 5
April 12 Day 30
April 15 Day 31
m&m’s
Section 7.1
April 17 Day 32
Confidence Intervals
Section 7.2
April 19 Day 33
Presentation 3
April 22 Day 34
Intro to Hypothesis Testing
Section 8.1
April 24 Day 35
Homework 7 due
Z-Procedures
Sections 7.3 and 8.2
April 26 Day 36
Testing Simulations
Section 8.4
April 29 Day 37
t-procedures
Section 8.2
May 1
More t-procedures
Section 8.2
May 3 Day 39
Homework 8 Due
Proportions
Sections 7.2 and 8.3
May 6 Day 40
Review
May 8
Homework 9 Due
Presentation 4
May 10 Day 42
Exam 2
Exam 3
Homework Assignments: (subject to change if we discover issues as we go)
Homework 1, due February 8
1)
Anxiety disorders and symptoms can often be effectively treated with medications. The
accompanying data on a receptor binding measure was read from a graph in a recent
scientific paper on the subject. Use various methods from Chapter 1 to describe and
summarize the data. In particular, we want to highlight the differences between the two
groups of patients.
PTSD:
2)
10
42
20
46
25
28
31
35
37
38
38
39
39
Healthy:
67
23
69
39
72
40
41
43
47
51
58
63
66
Consider a sample
and suppose that the values of
and s have been
calculated and are known. Let
and
for all i’s. (The y’s have been
“centered” and the z’s have been “standardized”.) Find the means and standard
deviations for the two new lists, y and z, in terms of and s.
3)
4)
Specimens of three different types of rope wire were selected, and the fatigue limit was
determined for each specimen. Construct a comparative box plot and a plot with all three
quantile plots superimposed. Comment on the information each display contains. Also
explain which graphical display you prefer for comparing these data sets. Keep in mind
your goal is to highlight the differences between the data sets.
Type 1
350
384
350
391
350
391
358
392
370
370
370
371
371
372
372
Type 2
350
380
354
383
359
388
363
392
365
368
369
371
373
374
376
Type 3
350
379
361
380
362
380
364
392
364
365
366
371
377
377
377
The sample data
sometimes represents a time series, where
is the observed
value of a response variable x at time t. Often the observed series shows a great deal of
random variation, which makes it difficult to study longer-term behavior. In such
situations, it is desirable to produce a smoothed version of the series. One technique for
doing so involves exponential smoothing. A smoothing constant α is chosen ( 0 < α < 1)
and then smoothed values
are calculated by
and for
.
a) Consider the following time series of the temperature of effluent at a sewage
treatment plant on day t: 47, 54, 53, 50, 46, 46, 47, 50, 51, 50, 46, 52, 50, 50. Plot
each x against t. Does there appear to be any pattern? Now calculate the ’s using
α = .1. Repeat for α = .5. Which value of α gives a smoother series?
b) Substitute
substitute
on the right-hand side of the expression for , then
in terms of
and
, and so on. On how many of the values
does
depend? What happens to the coefficient on
as k increases?
If t is large, how sensitive is
to the initial condition
?
Homework 2, due February 20
1)
A utility company offers a lifeline rate to any household whose electricity usage falls
below 240 kWh during a particular month. Let A denote the event that a randomly
selected household in a certain community does not exceed the lifeline usage during
January, and let B be the analogous event for the month of July (A and B refer to the same
household). Suppose
,
, and
. Compute a)
,
and b) the probability that the lifeline usage amount is exceeded in exactly one of the two
months. Describe this last event in terms of A and B.
2)
The route used by a motorist to get to work has two stoplights. The probability of signal
1 being red is .4 and for signal 2 it is .5. There is a .6 probability that at least one of the
two is red. What is the probability that both signals are red? That the first is red, but not
the second? That exactly one signal is red?
3)
Three molecules of type A, three of type B, three of type C, and three of type D are to be
linked together to form a chain molecule. An example of one such chain molecule is
ABCDABCDABCD and another is BCDDAAABDBCC.
a)
How many such chain molecules are there? [Hint: If the three A’s were
distinguishable from one another, such as
,
, and
, how many molecules
would there be? How is this number reduced when the subscripts are removed from
the A’s?]
b) Suppose a chain molecule of the type described is randomly selected. What is the
probability that all three molecules of each type end up next to one another (such as in
BBBAAADDDCCC)?
4)
Three married couples have purchased theater tickets and are seated in a row consisting
of just six seats. If they take their seats in a completely random fashion (random order),
what is the probability that Jim and Paula (husband and wife) sit in the two seats on the
far left? What is the probability that Jim and Paula end up sitting next to one another?
What is the probability that at least one of the wives ends up sitting next to her husband?
[Note: you probably won’t be able to use a formula to answer this; rather, a brute force
listing of the sample space may be a more fruitful strategy.]
Homework 3, due February 27
1)
In a Little League baseball game, suppose the pitcher has a 50 % chance of throwing a
strike and a 50 % chance of throwing a ball, and that successive pitches are independent
of one another. Knowing this, the opposing team manager has instructed his hitters to not
swing at anything. What is the chance that the batter walks on four pitches? What is the
chance that the batter walks on the sixth pitch? What is the chance that the batter walks
(not necessarily on four pitches)? Note: in baseball, if a batter gets three strikes he is out,
and if he gets four balls he walks.
2)
A car insurance company classifies each driver as good risk, medium risk, or poor risk.
Of their current customers, 30 % are good risks, 50 % are medium risks, and 20 % are
poor risks. In any given year, the chance that a driver will have at least one citation is 10
% for good risk drivers, 30 % for medium risk drivers, and 50 % for poor risk drivers. If
a randomly selected driver insured by this company has at least one citation during the
next year, what is the chance that the driver was a good risk? A medium risk?
3)
An insurance company offers its policyholders a number of different premium payment
options. For a randomly selected policyholder, let X = the number of months between
successive payments. The cdf of X is:
What is the pmf of X? Using just the cdf, compute P(3 ≤ X ≤ 6) and P(4 ≤ X). [Of course
you can use your pmf to check your work.]
4)
The pmf for X, the number of major defects on a randomly selected appliance in our
warehouse, is
Compute E(X), V(X) using the definition, and V(X) using the shortcut.
Homework 4, due March 15
1)
The time it takes a read/write head to locate a desired record on a computer disk once
positioned on the right track can be reasonably modeled with a uniform distribution. If
the disk rotates once every 25 msec, then assume
. Compute P(10 ≤ X ≤
20), P(10 ≤ X), the cdf F(X), E(X), and V(X).
2)
Use the following pdf and find a) the cdf b) the mean and c) the median of the
distribution.
.
3)
There are two machines available for cutting corks intended for use in wine bottles. The
first produces corks with diameters that are normally distributed with mean 3 cm and
standard deviation .1 cm. The second machine produces corks with diameters that have a
normal distribution with mean 3.04 cm and standard deviation .02 cm. Acceptable corks
have diameters between 2.9 cm and 3.1 cm. Which machine is more likely to produce an
acceptable cork?
4)
Suppose the time it takes for Jed to mow his lawn can be modeled with a gamma
distribution using α = 2 and β = 0.5. What is the chance that it takes at most 1 hour for
Jed to mow his lawn? At least 2 hours? Between 0.5 and 1.5 hours?
Homework 5, due March 29
1)
The data below are precipitation values during March over a 30-year period in
Minneapolis-St. Paul.
0.77
1.20
3.00
1.62
2.81
2.48
1.74
0.47
3.09
1.31
1.87
0.96
0.81
1.43
1.51
0.32
1.18
1.89
1.20
3.37
2.10
0.59
1.35
0.90
1.95
2.20
0.52
0.81
4.75
2.05
Construct and interpret a normal probability plot for this data set. The large outliers
should make the data look non-normal. Can a transformation make the data more normal
looking? Calculate the square root and the cube root of each observation, and construct
and interpret normal probability plots. What do you conclude is the best choice: leave the
data along, use the square root, or use the cube root?
2)
A particular type of tennis racket comes in a midsize version and an oversize version.
Sixty percent of all customers who shop at a certain store want the oversize version.
Assume the next ten customers that come to the store are a random sample of all
customers. (This assumption is sometimes difficult to justify in practice, but is almost
always made to facilitate our calculations. Whether this is good practice or not is a
worthy discussion.) What is the chance that at least six of the next ten customers want
the oversize racket? What is the chance that the number of customers out of the next ten
who want the oversize racket is within one standard deviation of the mean? If the store
currently has only seven rackets of each version, what is the chance that all of the next
ten customers can get the version they want? [Hint: It might be easier to calculate the last
part by examining the complement.]
3)
Suppose n in a binomial experiment is known and fixed. Are there any values of p for
which the variance is zero? Explain this result in words. For what value of p is the
variance maximized? [Hint: Either graph variance as a function of p or try to minimize
using calculus.]
Homework 6, due April 8
1)
A second stage smog alert has been called in a certain area of Los Angeles county in
which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to
check for violations of regulations. If 15 of the firms are actually violating at least one
regulation, what is the pmf of the number of firms visited by the inspector that are in
violation of at least one regulation? Find the Expected Value and Variance for your pmf.
2)
A couple wants to have exactly two girls and they will have children until they have two
girls. What is the chance that they have x boys? What is the chance they have 4 children
altogether? How many children would you expect this couple to have? (Find the
Expected Value.)
3)
Let X have a binomial distribution with n = 25. For p = 0.5, 0.6, and 0.9, calculate the
following probabilities both exactly and with the normal approximation to the binomial.
a) P(15 ≤ X ≤ 20). b) P(X ≤ 15). c) P(20 ≤ X). Comment on the accuracy of the normal
approximation for these parameter choices.
Homework 7, due April 24
1)
There are 40 students in a statistics class, and from past experience, the instructor knows
that grading exams will take an average of 6 minutes, with a standard deviation of 6
minutes. If grading times are independent of one another, and the instructor begins
grading at 5:50 p.m., what is the chance that grading will be done before the 10 p.m.
news begins?
2)
A student has a class that is supposed to end at 9:00 a.m. and another that is supposed to
begin at 9:10 a.m. Suppose the actual ending time of the first class is normally
distributed with mean 9:02 and standard deviation 1.5 minutes. Suppose the starting time
of the second class is also normally distributed, with mean 9:10 and standard deviation 1
minute. Suppose also that the time it takes to walk between the classes is a normally
distributed random variable with mean 6 minutes and standard deviation 1 minute. If we
assume independence between all three variables, what is the chance the student makes it
to the second class before the lecture begins? [Hint: Consider the linear combination
. Positive values of
correspond to making it to class on time.]
3)
A 90 % confidence interval for the true average IQ of a group of 100 people is (114.4,
115.6). Deduce the sample mean and population standard deviation used to calculate this
interval, and then produce a 99 % interval from the same data.
4)
An experimenter would like to construct a 99% confidence interval with a length of no
more than 0.2 ohms, for the average resistance of a segment of copper cable of a certain
length. If the experimenter is willing to assume that the true standard deviation is no
larger than 0.15 ohms, what sample size would you recommend?
Homework 8, due May 3
1)
Fifteen samples of soil were tested for the presence of a compound, yielding these data
values: 26.7, 25.8, 24.0, 24.9, 26.4, 25.9, 24.4, 21.7, 24.1, 25.9, 27.3, 26.9, 27.3, 24.8,
23.6. Is it plausible that these data came from a normal curve? Support your answer.
Now calculate a 95% confidence interval for the true average amount of compound
present. Comment on any assumptions you had to make.
2)
A hot tub manufacturer advertises that with its heating equipment, a temperature of
100°F can be achieved in at most 15 minutes. A random sample of 32 tubs is selected,
and the time necessary to achieve 100°F is determined for each tub. The sample average
time and sample standard deviation are 17.5 minutes and 2.2 minutes, respectively. Does
this data cast doubt on the company’s claim? Calculate a P-value, and comment on any
assumptions you had to make.
3)
A sample of 50 lenses used in eyeglasses yields a sample mean thickness of 3.05 mm and
a population standard deviation of .30 mm. The desired true average thickness of such
lenses is 3.20 mm. Does the data strongly suggest that the true average thickness of such
lenses is undesirable? Use α = .05. Now suppose the experimenter wished the
probability of a Type II error to be .05 when µ = 3.00. Was a sample of size 50
unnecessarily large?
4)
Suppose that the true average viscosity should be 3000 in a certain process. Do the
following measurements support that standard? State and test the appropriate hypotheses.
2781
2900
3013
2856
2888
Homework 9, due May 8
1)
A random-number generator is supposed to produce a sequence of 0s and 1s with each
value being equally likely to be a 0 or a 1 and with all values being independent. In an
examination of the random-number generator, a sequence of 50,000 values is obtained of
which 25,264 are 0s.
a) Formulate a set of hypotheses to test whether there is any evidence that the randomnumber generator is producing 0s and 1s with unequal probabilities, and calculate the
corresponding P-value.
b) Compute a 99% confidence interval for the probability p that a value produced by the
random-number generator is a 0.
c) If a two-sided 99% confidence interval for this probability is required with a total
length no larger than 0.005, how many additional values need to be investigated?
2)
In a survey of 4,722 American youngsters, 15 % were seriously overweight, as measured
by BMI. Calculate and interpret a 99 % confidence interval for the proportion of all
American youngsters who are seriously overweight. Discuss whether the Associated
Press (who reported this data) actually took or could have taken a random sample of
American youngsters.
3)
In is known that roughly 2/3 of all human beings have a dominant right foot or eye. Is
there also right-sided dominance in kissing behavior? One scientific article reported that
in a random sample of 124 kissing couples, both people in 80 of the couple tended to lean
more to the right than to the left. If 2/3 of all kissing couples exhibit this right-leaning
behavior, what is the probability that the number in a sample of 124 who do so differs
from the expected value by at least as much as what was actually observed? (i.e.
calculate a P-value.) Does the result of the experiment suggest that the 2/3 figure is
plausible or implausible? State and test the appropriate hypotheses.
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Last updated January 18, 2013