The death of one man is a tragedy. The death of a million men--that is statistics. --Joseph Stalin
Office Hours: | Tuesday 4:15-5:15 or by appointment |
Class Meetings: | Tuesdays and Thursdays, 2:00-3:50, X-hour Wednesday 4:15-5:05, Room 102 Bradley Hall |
Texts: | Statistics, Third Edition, by Freedman, Pisani, and Purves |
ActivStats CD-ROM by Paul Velleman |
Class 1 9/30 | Class 2 10/2 | Class 3 10/7 | Class 4 10/9 |
Class 5 10/14 | Class 6 10/16 | Class 7 10/21 | Class 8 10/23 |
Class 9 10/29 x-hour | Class 10 10/30 | Class 11 11/4 | Class 12 11/6 |
Class 13 11/11 | Class 14 11/13 | Class 15 11/18 | Class 16 11/20 |
Class 17 11/25 | 11/27 Thanksgiving | Class 18 12/2 | Class 19 12/4 pre-exam break |
Topics that might be covered in Chance include:
Discussions are central to the course and usually focus on a current article in the news. They provide a context in which to explore questions in more depth and understand material better by explaining it to others. Due to the interactive nature of the course, you will be expected to come to class and to engage whole-heartedly in the discussions. Every member of each group is expected to take part in these discussions. Each of you also have a responsibility to make sure that everyone is involved, that everyone is being heard, everyone is listening, that the discussion is not dominated by one person, that everyone understands what is going on, and that the group sticks to the subject.
Each participant should keep a journal for the course, which is separate from the weekly homework assignments. A good journal should answer questions asked in class and should include
In writing in your journal, exposition is important. If you are presenting the answer to a question, explain what the question is. If you are giving an argument, explain what the point is before you launch into it. What you should aim for is something that could communicate to a friend or a colleague a coherent idea of what you have been thinking and doing in the course.
We encourage you to cooperate with each other in working on anything in the course, but what you put in your journal should be your own alone. If you include something that has emerged from work with other people, write down who you have worked with. Ideas that come from other people should be given proper attribution. If you have referred to sources other than the texts for the course, cite them.
Your journal should be kept on loose-leaf paper. We will collect journals periodically to be read and commented on. If they are on loose-leaf paper, you can hand in the parts that have not yet been read, and continue to work on further entries. Pages should be numbered consecutively and, except when otherwise instructed, you should hand in only those pages that have not previously been read. Write your name on each page and, in the upper right hand corner of the first page you hand in each time, list the pages you have handed in (e.g. [7,12] on page 7 will indicate that you have handed in 6 pages numbered 7 to 12).
Journals will be collected and read roughly every two weeks. Tentative dates are as follows:
Thursday, October 9 | Thursday, November 21 |
Thursday, October 23 | Tuesday, December 2 |
Thursday, November 6 |
To supplement the discussion in class and assignments to be written about in your journals, we will assign readings from Freedman-Pisani-Purves, interactive investigations from the ActivStats CD-ROM, as well as accompanying written homework. When you write the solutions to these homework problems, you should keep them separate from your journals. Homework will be assigned once a week and should be handed in on Thursdays.
We will not have a final exam for the course, but in its place, you will undertake a major project. This project may be a paper investigating more deeply some topic we touch on lightly in class. Alternatively, you could design and carry out your own study. Or you might choose to do a computer-based project. To give you some ideas, a list of possible projects will be circulated. You can also look at some previous projects on the Chance Database. However, you are also encouraged to come up with your own ideas for projects.
At the end of the course we will hold a Chance Fair, where you will have a chance to present your project to the class as a whole, and to demonstrate your mastery of applied probability by playing various games of chance. The Fair will be held during the final examination time assigned by the registrar.
Your grade will be determined by your homework, including Freedman-Pisani-Purves and computer assignments, journals and class participation, and final project. There may also be a test.
Materials related to the course will be placed on the World Wide Web at the address http://math.dartmouth.edu/~math5/section2. In addition supplementary readings will be kept on reserve in Baker Library.
Discussion Questions
Apply Tough Standards To Both Tranditional And Alternative Medicine
Discussion Questions
The placebo question: Can research kill?
The Boston Globe, Sept.18, 1997
By Richard A. Knox
1. What is the answer to the question posed in the first paragraph? (Should some babies be denied a drug known to prevent HIV infections in a study to see if cheaper treatments are equally effective?)
2. What is the justification for the use of the placebo in these studies? Why might it take longer to obtain results if a placebo is not used?
3. Dr. Marcia Angel believes that "Only when there is no known effective treatment is it ethical to compare a potential new treatment with a placebo." Do you agree?
4. Federal officials say that using a placebo is not depriving women of therapy they would have otherwise received since AZT is not affordable in the Third World. How would you respond to this statement?
5. How should this study be conducted?
Survival of AIDS Patients Linked To Experience
of Their Doctors
The New York Times, Feb. 1, 1996
Lawrence K. Altman
1. Why did this study use the median survival time rather than the mean for comparing the patients of different doctors?
2. What kinds of confounding factors may be present in this study?
3. The articles says that "researchers took into account...the severity of illness" of the patients in the study. What does this mean?
4. What is convincing in this article? What isn't?
Ask Marilyn
Parade Magazine, 20 March 1994, p. 31
Marilyn vos Savant
A reader wonders why the probability of rain is not always 50% since there are only two outcomes. Marilyn points out the absurdity of this, commenting that this kind of logic would lead to a 50% chance that the sun will not rise tomorrow. She goes on to say:
But rain doesn't obey the laws of chance; instead, it obeys the laws of science. It would be far more accurate for a meteorologist to announce, "There's a 25% chance that a forecast of rain will be correct."
DISCUSSION QUESTIONS:
(a) What do you make of all this?
(b) What does Marilyn mean when she says, "But rain doesn't obey the laws of chance; instead it obeys the laws of science"? Do you agree with her?
(c) If a forecaster says the probability of precipitation (POP) is 30% and it rains, is the forecaster correct?
(d) Suppose that Hanover gets precipitation 2 days out of 10 over the long run. Why not report a POP of 20% every day?
(e) How does your local weather forecaster decide on the probability for rain tomorrow? (Incidentally, if anyone REALLY knows how this is done we would love to hear the answer).
(f) Suppose you have two forecasters, each of whom give a POP every day. If you were running a contest to reward the best forecaster, how would you decide the winner?
Behind Monty Hall's Doors: Puzzle, Debate and Answer?
The New York Times, July 21, 1991
John Tierney
A mammogram is an x-ray used in the early detection of breast cancer. 1% of all people receiving mammograms actually have cancer. If a person has cancer, then a radiologist will correctly diagnose it as cancer 80% if the time. (I.e. the probability that a person with cancer will receive a positive result on a mammogram is .8.) If a person doesn't have cancer, then a radiologist will incorrectly diagnose it as cancer 10% of the time. (I.e. the probability that a person without cancer will receive a positive result is .1.)
Given this, suppose you have a mammogram and receive a positive result. What is the probability that you actually have cancer?
1. Take a guess. What is your intuitive answer to this question? ________
2. Try working the answer out with a tree diagram. If this gets confusing, move on to 3.
3. Work this out by constructing a tree diagram with number2 of people rather than probabilities:
1. Suppose 1000 people have mammograms.Now the probability that a person has cancer given that they have a positive test result is2. How many of these people actually do have cancer? ________
3. How many don't have cancer? ________
4. Of those that do have cancer, how many will receive a positive result? _______
5. A negative result? ________
6. Of those that don't have cancer, how many will receive a positive result? _____
7. A negative result? ________
number of people with cancer and a positive resultnumber of people with a positive result
What is this probability? ________
Make sure every member of the group understands where these numbers come from before you go on. Ask your instructors for help if necessary!
4. Translate this problem into probability language as follows:
Let C be the event that a person has cancer.What is P(C)? ________
Let N be the event that a person doesn't have cancer.
Let + be the event that a person has a positive result on a mammogram.
Let - be the event that a person has a positive result on a mammogram.
5. Now try to do 2 again. First fill in the mathematical symbols and then fill in their values.
6. We can see from the diagrams that
This is a famous formula with a special name. It is called Bayes formula. It is used when one has two events A and B (in our case C and N) that are mutually exclusive and account for all cases. (In our example one can't have cancer and not have cancer simultaneously. The events C and N are therefore mutually exclusive. All cases are accounted for since every person either has cancer or doesn't have cancer.) We use the formula when we have a third event E (in our case +) and we want to know P(A | E) but only have information concerning P(E | A). (How does this statement relate to our example?)
In fact we can use this formula for any number of mutually exclusive events that account for all cases. Let the events be A1, A2, ... , An where n is any integer. Let E be an event. Then Bayes formula says
P(E | A1)P(A1) + P(E | A2)P(A2) + ... + P(E | An)P(An)
where i is any number between 1 and n.
The standard procedure for HIV testing is as follows: first an ELISA test is performed. If the test is positive, a second ELISA test is performed. If the second test is positive, then a third test, called the Western blot test is performed. If all three tests are positive, a positive result is reported.
The combined ELISA/Western blot test has a sensitivity of 99.8%, meaning that it returns a positive result for 99.8% of infected persons. The ELISA/Western blot test has a specificity of 99.9%, meaning that it returns a negative result for 99.9% of non-infected persons. Thus, the false-positive rate is 0.1% and the false negative rate was about 0.2%.
In 1992 the US Centers for Disease Control estimated that the HIV infection rate was 1.5 in 10,000 for the population classified as "heterosexuals without specific identified risk". This population comprises an estimated 142 million Americans. If a person from the "heterosexual without specific identified risk" tests positive, what is the chance this person has the HIV virus?
What would this probability be for a member of a high risk population in which 5% of the population has the HIV virus?
Would you recommend that all adults be routinely tested for AIDS? If you ran the Food and Drug Administration, how would your answers affect your decision on approving or disapproving a home AIDS test?
In calculating the probability that a person who tests positive on the set of three tests has the disease, is it reasonable to assume that these three tests are independent chance experiments?
A recent article in the New York Times reports an HIV infection rate for college students of 2 in 1,000. A 1992 study funded by the National Institute of Allergy and Infectious Diseases estimates the risk of heterosexual transmission of HIV from a single sexual encounter as follows:
Male-to-female transmission rate: | 0.0005 |
Female-to-male transmission rate: | 0.00005 |
(These estimates are for unprotected sex) |
(For the following problems, you may choose the uninfected student to be of either gender)
If an uninfected student has a single encounter with a randomly selected student of the opposite sex, what is the probability that he/she will be infected with HIV?
If an uninfected student has 1 encounter each with 100 randomly selected students of the opposite sex, what is the probability that he/she will be infected with HIV?
If an uninfected student has 100 encounters with one randomly selected student of the opposite sex, what is the probability that he/she will be infected with HIV?
How do you explain these results?
Is the above procedure a reasonable way to estimate risk? If not, why not? If so, what are the implications?
Parental age gap skews child sex ratio
Nature Vol 389, 25 Sept. 1997, p. 344
J. T. Manning, R. H. Anderton, M. Shutt
A run on baby girls at Noyes.
Livingston County (NY) News, 21 August 1997, p.1
Ben Carlson:
Heads or tails, boy or girl, the odds are the same. But earlier this month, Nicholas H. Noyes hospital in Dansville had an unusual run of births as a string of 12-straight girls were born in a row. According to Nurse Manager Amy Nasca, it wasn't something she and her staff were actively keeping track of, but she said it was very interesting to watch such an unusual trend develop. "We didn't realize we had so many girls in a row until we actually counted them up," said Nasca. "It would have been fun to make it to 13 in a row, but what we were actually more concerned about was having healthy babies be born." With the normal odds running at 50-50 to have either a boy or girl, the odds of having 12 of either sex in a row are 4,096-to-1. But according to Nasca, things have a way of evening out. "Earlier this year, a string of 11 baby boys were born," she said. "(But) it seems by the end of the year our statistics always even out." According to hospital statistics, of the 168 babies born so far this year, 80 have been girls. "We do have times where we will have several boys or girls in a row," said Nasca. "It's fun, but we are really just happy to have healthy babies born here."
DISCUSSION QUESTIONS:
(1) Is the statement "the odds of having 12 of either sex in a row are 4,096-to-1" accurate.
(2) How would you find the chance that, of the 168 babies born so far this year, 80 are girls?
(3) A student answers question 2 by: the probability is 1 since it has already happened. Is she correct?
(4) About how many times do you think you would have to toss a coin to get 12 heads in a row?
(5) Many people believe that families "tend to have boys" or "tend to have girls", so the coin tossing model would not be appropriate. Do you think there is any basis for this?
Use of Placebos in Ivory Coast AIDS Test Sparks Ethics Debate in U.S.
The New York Times
by Howard W. French
Higher speeds mean more deaths on U. S. roads
Nando Times News (October 10, 1997 http://www.nando.net)
Condom distribution not linked to increase in sex, study says
Nando Times News (September 30, 1997 http://www.nando.net)
Common Type of Fat linked to Breast Cancer
The New York Times, October 14, 1997
by Jane E. Brody
Remember the survey you completed in the first week of the term? We have entered your responses into the computer and now it is time to have a look at the data. This sheet introduces three ways of displaying data: histograms, scatterplots, and boxplots. The CD assignment for next week covers the details of these kinds of displays. This sheet serves as a quick introduction to allow you to start looking at data today.
The data is available on PUBLIC. If you know where PUBLIC is, then open it now. Otherwise do File-->Find and type in PUBLIC. When you have PUBLIC open, click on the following sequence of folders: Courses & Support-->Academic Departments & Courses-->Math-->Chance-->97Chance. Drag the 97 survey data icon to the desktop and close everything else. Double click on the icon to open data desk.
With data desk open you will see a row of variables. To see what's in one, double click on it. Close the icon by clicking on the little stop sign shape in the upper left corner.
To make a histogram, select a variable by clicking on it. Do Plot-->Histograms. Try this with the Height variable. To do separate histograms for different classes, select the Class variable and do Special-->Group-->Assign. A bunon will appear on the bottom of the screen. With the button on, select the variable Height again. Do Plot-->Histograms and you will see one histogram for each class. Try this again with other variables of your choice.
Now plot a histogram of the Birthorder variable. In the upper left corner of the histogram is a little triangle. Click on this and drag down to Plot Scale... Change the bar width to 1. Change it to 2, then 3. What is happening? Change the bar width back to 1. Does the information displayed in this graph surprise you?
To make a scatterplot, select the variable you are interested in knowing about by clicking on it. Next select the variable that you think might explain the behavior of the first by pressing the shift button and then clicking on the second variable while the shift button is still held down. Do Plot-->Scatterplots. For example suppose you want to investigate Verbal SAT scores. Click on the VSAT variable. Suppose you conjecture that higher Math SAT scores will be associated with higher Verbal SAT scores. Select MSAT as the second variable by holding down the shift button and then clicking on the variable MSAT. Do Plot-->Scatterplots. Do you see an association?
Marilyn vos Savant has a bet with one of her readers about the following problem: "A woman and a man (unrelated) each have two children. At least one of the woman's children is a boy, and the man's older child is a boy. Do the chances that the woman has two boys equaal the chances that the man has two boys?" Marilyn has bet $1000 that the chances that the man has two boys are higher than that the woman has two boys. Why?
First proof: driving while talking on phone is a hazard.
Association Between Cellular-Telephone Calls and Motor Vehicle Collisions
The New England Journal of Medicine, Vol. 336, No. 7 p. 453, Feb. 13, 1997
Donald A. Redelmeier and Robert J. Tibshirani
Is using a car phone like driving drunk?
CHANCE Vol.10 No.2, 1997
Donald A. Redelmeier and Robert J. Tibshirani
1. How was the study designed? How do the authors estimate the relative risks of driving while using/not using a car phone?
2. What are the major confounding variables that the authors try to account for?
3. Do you think the authors should be surprised at the media's interpretation of their statement in the NEJM article that compared the risk of using a cellular phone while driving to that of having a blood alcohol level corresponding to being legally impaired? Why do you think the authors made the companson in the NEJM article?
4. Writing about an earlier study carried out in 1978, the authors say: "This survey of 498 individuals found that the overall frequency of traffic violations was marginally lower among the mobile telephone subscribers than among members of the general public (11% vs. 12%)." Why do you think the authors were suspicious of the results of this survey?
5. Writing about another study carried out in 1985, the authors say: "This study of 305 individuals found a significantly lower collision rate in the year following the purchase of a cellular telephone (8.2% vs. 6.6%)." They were "impressed" by this study but also "worried". Why?
What makes one chocolate chip cookie taste better than another? Today we will attempt to answer this question by means of an experiment. Before we dive into the cookies, we need to decide exactly what we're testing for, and how we expect to measure the response.
Step 1. What could possibly make one cookie taste better than another? In what
ways do the cookies vary? With your group write down all possible factors that differentiate cookies. Consider even those factors which you expect to be irrelevant (size, for example).Step 2. Decide how to measure each of the different factors you mentioned above. For example, does size mean diameter, thickness, weight, or volume? Or perhaps you could invent your own measure of size. What would it be? Should cookies be compared individually to some predetermined scale, or should they be placed in order from best to worst, or both?
At this point pause to confer with the rest of the class. We will decide as a class what factors we want to consider so that we can pool our data.
Step 3. Design the experiment. Will every person test every cookie for every factor? What order should the cookies be presented in? Should the testers know which cookie is which brand before they start testing? Should the cookies be present whole or crumbled? If a cookie is being tested for a factor that doesn't involve appearance, should the tester be able to see the cookie? What other design choices need to be made?
Pause again to get an agreement with the rest of the class on experiment design.
Sbp 4. Perform the experiment. Good luck!
House May Seek Court Ruling on Census Sampling
New York Times on the web, Aug. 30, 1997
David Stout
DISCUSSION QUESTIONS:
1. Does the proposed plan to use sampling in the 2000 census account for everybody? Does this plan "correct the kind of undercount that plagued the 1990 census"? Who do you think was missed in 1990?
2. As a final "quality control" step in the proposed plan, the census bureau wants to "randomly select 750,000 households nationwide, making sure that each state and all racial, age, economic and other demographic groups were represented." If the sample is random, how can they "make sure" that all groups are represented?
3. Do you think the proposed method is more or less accurate than the existing method of enumeration? Is sampling constitutional?
Final projects are due the day of the Chance Fair which will occur sometime during the reading period.
Projects should be done individually or in groups of two. In the latter case it should be clear who did what and the project should be twice as involved as a project completed by one person.
The main point of the project is to convince the professors that you learned something in this course. We need to assign grades and we are offering you the opportunity to prove to us that you deserve a good one.
Your final project need not be an experiment, but to give an example of what the professors are looking for in a project we will consider the cookie experiment performed in class.
Many people pointed out design flaws in the cookie experiment. Many of these flaws resulted from a lack of time to carefully plan and perform the experiment. Donıt let this happen to you with your project! Give yourself plenty of time to work through the details and think about possible confounding in advance. Also be warned that people are busy towards the end of the term. If you plan to use Dartmouth students in a survey or an experiment, it may be harder to get volunteers at the last minute.
In the cookie experiment the professors would look for the following in a final project:
Question or Hypothesis. There are many questions to ask about cookies. What makes them taste good? How does a cookieıs brand influence a customerıs intent to purchase? Do men and women have different cookie preferences? Et cetera. Choose a question to answer. The question asked or the claim made should be clear to the professors. Also consider about whom you are asking the questions. What makes cookies taste good to whom? To the U.S. population? Hanover citizens? Dartmouth students? Mass Row residents?
Factors Considered. What information do you need to answer your question? For example you may choose to consider size, chewiness, and fat content of cookies. The professors want to see that you have chosen reasonable factors and that you have considered everything relevant. If you donıt consider chocolate content of chocolate chip cookies, for example, the professors may wonder why. Donıt hesitate to consider factors that may at first seem irrelevant, such as size of cookies. If there are obvious ways to distinguish the things you test, you might as well include them just to see what happens.
How Factors are Measured. If you considered size of cookies, you would have to explain how you measured size. Did you measure thickness, diameter, weight, volume, or something else? Why did you choose this way of measurement? Justify your choices by saying why you thought this would be the most accurate or meaningful measurement.
Data Interpretation. This is where you plug all of your data into data desk and try to make some conclusions. Did you consider all reasonable correlations? How did you deal with outliers?
Conclusions. What did your data tell you? What is the answer to your question? Did your data support or negate your hypothesis? Keep in mind that a negative result is not bad. Perhaps the answer is that there is no correlation between gender and cookie preference. If this is what you found, then thatıs fine. You donıt need a positive result to have done a good project. Think about possible confounding factors in your study. What would you do differently if you had to do your project again?
Study Links Rate of Violence to Cohesion in Community
New York Times on the web, Aug. 17, 1997
Fox Butterfield
Note:
For the crack vs. homicide articles, we split the class up into two
teams: the statisticians and the police force. We also had a jury of
five people. It was the statisticians' job to convince the jury that
the rate of use of crack influences the homicide rate. The police force
tried to convince the jury that there are factors besides the use of
crack that influence the homicide rate, for example better policing.
The jury decided both the format of the debate and made a verdit in the
end.
Woman March on Philadelphia"
The Dartmouth, Oct. 27, 1997
The Associated Press
Note:
We talked about how one could estimate such a thing as the number of women in
Philadelphia one day.
DISCUSSION QUESTIONS:
1. What do you think of the two different methodologies used in conducting the surveys? What are good and bad aspects of each?
2. How might you compare the two methods more directly?
3. If you had to estimate drug use in schools, how would you go about doing so?
GOP Proposes Mailing IRS Customer Satisfaction Survey with Tax Forms
LEGI-SLATE News Service, News of The Day (10/31/97)
Sara Hebel
DISCUSSION QUESTIONS:
1. What are some problems with the methodology used in the proposed suevey?
2. How might you design things if your goal was to get as accurate a picture as possible of customer satisfaction with the IRS?
3. Suppose you are a strong supporter of government-run universal health care and that you want a poll showing that the American public strongly supports the idea of universal health care. How might you design a survey question?
4. Suppose you are strongly opposed to government-run universal health care and that you want a poll showing that the American public strongly opposes the idea of universal health care. How might you design a survey question?
Straights, gays lament dating scene, D-Plan, Greeks
The Dartmouth, Feb. 14, 1997
Erin Loback
Majority of campus claims sexual activity, 11% daily
The Dartmouth, Feb. 14, 1997
Erin Loback
DISCUSSION QUESTIONS:
1. Do you think the results of the survey are reflective of campus-wide levels of sexual activity? Why or why not?
2. How would you design a study to get the most accurate results possible?
Confidence Intervals Using M&M's
What is the fraction of green M&M's in a bag of M&M's? How many green M&M's can we expect when we buy a small bag? What range of answers do we expect to this question? How sure are we of the answers we arrive at? How many M&M's do we need to sample to get a good estimate? This activity examines the answers to these questions.
What percentage would you guess for the green M&M's? ________
If we know the percentage of green M&M's produced, then we can figure out how many to expect in a small bag of fifty M&M's. Suppose 10% of all M&M's made are green. Set up a box model.
1. Each ticket in the box represents what?
2. Are we interested in the sum of the numbers on the tickets drawn, or something else? What does the sum (or something else) tell us?
3. How is each ticket labeled? Why?
4. How many tickets should be drawn from the box?
Now that we have the box model, what is the SD for the box? Verify that both definitions for the SD that we have been using give the same answer. What is the SE for the fifty draws from the box?
Drawing fifty tickets from the box and taking the sum is like opening a small bag of fifty M&M's and counting the number of green candies. The numbers you get should be exactly the same. Why?
Fill in the blanks:
Assuming 10% of all M&M's are green, in a small bag of fifty candies, you would expect to get __________ green candies plus or minus __________ candies 68% of the time. You would expect to get __________ green candies plus or minus __________ candies 95% of the time. In other words, you would expect to see __________ percent green plus or minus __________ percent 68% of the time. You would expect to get __________ percent green plus or minus __________ percent 95% of the time.Now suppose you're in the following, more realistic situation. You don't know the percentage of green M&M's produced, yet you still want to figure out how many green candies to expect in a small bag of fifty. If you know the percentage of green M&M's, then you can figure everything out as above. This time you don't know the percentage, so youıll have to estimate it by sampling. Do so now.
Estimated percentage of green M&M's: __________Using the same box model as above with the new percentage of green M&M's estimated from your sample, figure out the SD for the box and the SE for a sample of fifty.For possible future use, please record the following as well:
% brown_____ % red_____ % orange_____ % yellow_____ % blue_____
SD for the box: __________Use your new numbers to fill in the blanks:SE for a sample of fifty: __________
Assuming __________% of all M&Mıs are green, in a small bag of fifty candies, you would expect to get __________ green candies plus or minus __________ candies 68% of the time. You would expect to get __________ green candies plus or minus __________ candies of the time. In other words, you would expect to see __________ percent green plus or minus __________ percent 68% of the time. You would expect to get __________ percent green plus or minus __________ percent 95% of the time.Note that the range of values expected 95% of the time chances as the expected number or percent of green M&M's changes. To see this, compare your results with those of others in the class.
Are you satisfied with the range of possible answers above? Does the plus or minus range seem unsatisfactory? How many M&M's do you need in a sample to get the accuracy you want 95% of the time? (For those of you doing surveys for your projects, this is how you figure out how many people to survey.)
What do you want your plus or minus range to be in terms of percents?
__________This number will be called X in the discussion below.
Now X comes from taking the size of 2 SE's and dividing by the size of the sample. Since we are trying to find the size of the sample, we will call it N. For you, the variable X is an actual number so we only have one unknown quantity N in what follows.
X = __________ (write X as a decimal here instead of a percent)
Now X = (2 x SE) / N = (2 x N x SD) / N. Collecting the N's on one side gives N / N = (2 x SD) / X . Since N / N = N , we have N = (2 x SD) / X. Squaring both sides yields N = (2 x SD) x (2 x SD) / (X x X). This means that if you want the percentage of green M&M's that you find in a sample to be accurate within the range plus or minus X = __________ percent 95% of the time, you need to take a sample of N = __________ M&M's.
Now that you know how to figure out how many M&M's to sample in order to get the accuracy you want when computing the percentage of green candies, how can you apply these techniques to the situation of surveying people? What do M&M's tell you about the number of people to survey to get the degree of accuracy you want?
A common method of estimating the number of fish in a lake or pond is the capture recapture method. In this method, c fish are caught, tagged, and returned to the lake. Later on, r fish are caught and checked for tags. Say t of them have tags. The numbers c, r, and t are used to estimate the fish population.
I) What is your estimate for the fish population in terms of c, r, and t? (It may help to think about actual numbers first.)2) What if some of the tags fall off your fish? Will your estimate be too big or too small?
3) Do you think that fish caught the first time are more likely to be caught the second time (or less likely to be caught)? If so, how would this affect your estimate?
4) What other assumptions do you need to make for your estimate to be reasonable?
5) The Census Bureau uses capture recapture to assess the number of people who were not counted by the Census ("the Census undercount"). Can you think how they might do this?
We will use capture/recapture to estimate the number of M&M's in a package.
The hidden truth about liberals and affirmative action
The Washington Post, 21 Sept. 1997, C5
Richard Morin
DISCUSSION QUESTIONS:
(1) Jim Baumgartner says this whole atudy upsets him. Why do you think he said this?
(2) Consider a particular group, say Democrats. Because the choice of who got three items and who got four was random, we can assume that about half the Democrats were given the three items and the other half were given the same three items plus the affirmative action item. Suppose that the average number of items that angered or upset the three-item group was found to be 1 and for the four-item group it was found to be 1.5. Then the authors would claim that about 50 percent of the Democrats must have said they were angry or upset about the affirmative action item. Why?
(3) When people indicate, in a poll like this, that they are angry about, for example, athletes getting million-dollar-plus salaries, do you think that they really are angry?
(4) Is it reasonable to assume that the response to the three items would be the same as they would be if they were asked together with a fourth item?
1. Can you tell from a student's answer if they have ever cheated? If
they have never cheated?
2. Suppose that there are no cheaters on campus. How many "Yes"
responses do you expect? How much variation do you expect due to chance?
3. Suppose that 65 of the 100 students you survey answer "Yes" to your
question. What percentage of students have actually cheated?
4. Based on your answer to question 3, how many of the 65 "Yes"
respondents have actually cheated? What is the probability that a "Yes"
respondent has actually cheated? How does this compare to the
probability that a randomly selected student has cheated?
5. You present your findings to President Freedman. The President, who
is convinced of the honesty of Dartmouth students, suggests that your
results may be due chance variation. How would you decide if he is
right?
6. Formulate a hypothesis test to answer question 5. What is your null
hypothesis? What is your alternative hypothesis?
7. What is the probability that you would obtain 65 or more "Yes"
answers if there were no cheaters on campus?
8. Are there any cheaters on campus, or are the results of your test due
to chance? Can you answer with certainty?