Teacher
education must be recognized as an important part of mathematics departments’ mission at institutions that educate teachers.
More mathematics faculty should consider becoming deeply involved in K-12 math education. The mathematics education of teachers
should be seen as a partnership between mathematicians and math educators.
The UGA Department of Statistics recognizes the
importance of providing the valuable service of training teachers to teach statistics at the K-12 level. UGA has a visionary
College of Education that recognizes what the teachers’ needs are in the area of statistics. The relationship between the two
departments has been successful because of mutual respect. The departments have worked together to produce a successful course that is in its fourth year of
being offered. Currently, the two departments are working together
to develop a similar course for K-8 teachers that will be taught in Spring 2002.
There is a need for collaborations between mathematics faculty and school math
teachers.
UGA Statistics faculty members have made frequent
classroom visits, giving guest lectures which helps teachers and students to see the relevance of statistics. Just as
importantly, in-service teachers have provided valuable perspectives for the instruction of teachers in the area of statistics.
These are teachers that are in the ‘trenches’ and they can give a real world perspective of what is needed in the classroom.
Efforts to
improve standards for school mathematics instruction will be strengthened by participation of the academic mathematics community.
Well respected statisticians have been actively
involved since the 1970’s with developing quantitative literacy through the development
of valuable statistical resources, AP Statistics, and working with
developing the NCTM Math Standards and the MET document. The American Statistical Association and the University of Georgia
are currently supporting and organizing a ‘Conference on Statistics in Teacher Preparation Programs’ to be held in
2003.
Where Do We Go from Here?
Quantitative literacy, also referred to as
numeracy, is the ability of an individual to make use of mathematical skills for problem solving in quantitative situations both
in everyday life and work.
Lynn A. Steen states in Education Week on the Web [Wednesday, September 5, 2001, Vol.21, No.1, p.58],
“…numeracy
is not the same as mathematics, nor is it an alternative to mathematics. Today’s students need both mathematics and
numeracy.
Whereas mathematics asks students to rise above context, quantitative literacy is anchored in real data that reflect engagement
with life’s diverse contexts and situations. The case for numeracy in schools is
not a call for more mathematics, nor even for more applied mathematics. It is a call for a different and more meaningful pedagogy
across the entire curriculum. In life, numbers are everywhere, and the responsibility for fostering quantitative literacy should
be spread broadly across the curriculum. Quantitative thought must be regarded as much more than an affair of the mathematics
classroom alone.
We must work hard to train our teachers to make
quantitative literacy and use of data as a main objective of K-12 education. An activity illustrating how we can attempt to train
our teachers by integrating content and pedagogy is presented in the appendix. This activity, written by the author of this paper,
will appear with full discussion in the upcoming NCTM Navigation Series, Data Analysis for the Secondary Level, scheduled to appear in print in April 2002. The activity is presented to the teachers, as
they would use the activity in their classrooms. After going through the activity, the teachers hopefully understand the concepts
being illustrated and the pedagogy being used to help students visualize the concepts. It is then that the mathematical theory
behind the activity can be presented to the teachers. In the past, teachers would have been presented with only the mathematical
theory.
Part II: Teaching the Extra-Mathematical Statistics Content
Robert Gould, UCLA, Department of Statistics, rgould@stat.ucla.edu
Although the intersection between Statistics and Mathematics is quite broad, there is sufficient
"extra-mathematical" content in Statistics to make it challenging to develop a curriculum that will adequately prepare a
teacher to teach both math and Statistics. I use "extra-mathematical" to refer to ideas and concepts that might be "math like", but
essentially have no basis in mathematics.
This discussion will be motivated by examination of a particular case-study.
Such an examination does not survive well in translation from a group discussion to a static paper, and those interested in
more details are encouraged to either consult the reference in which this case study is published
(Trumbo 2001), view the notes
from the MET workshop (www.stat.ucla.edu/~rgould/met), or -- best of all --
examine the data themselves at www.stat.ucla.edu/~rgould/met/lead.dat.
The data for this case study will be examined using Fathom, a software package designed for classroom
instruction. This study, while somewhat old (particularly by high-school
student standards) is none-the-less of current interest. In 1982, Morton et. al published a study examining lead absorption in the children of employees of a battery
factory in Oklahoma. Blood samples were taken from 33 children whose parents worked
around lead (the "Exposed group") and from 33 children whose parents did not work around lead (the "Control
group"). Ordinarily, when approach a data set, one should gather as much
information about the data as possible before beginning. But here, in order to
illustrate certain points, I'll hold onto some information and leak it a strategic moments.
Case Study
Our goal is to examine this data to see if there is evidence that the parents' work is inadvertently
increasing the lead content of their children's blood. High levels of lead in a
child's blood can cause severe developmental problems, and so is of great concern. Once
in the blood, the body has no mechanism for cleaning out the lead, and therefore lead levels increase over time with repeated
exposure.
A common approach in "chug-and-plug" statistics is to compute the average of the sample. Of course
we'll learn nothing from considering on ly the average of the Exposed group; we must compare it to the average of the Control
group.
We can ask Fathom to compute these, and we learn that the average lead level in the Exposed children is
approximately 32 micrograms per deciliter (mg/dcl), and in the Control group it is 16 mg/dcl.
Now no one contests that 32 is a bigger number than 16, but the real question is does it matter? There are (at least) two ways of approaching this. One
requires the input from medical specialists. They would tell us that levels of 40 and
above are considered alarming, and some experts say levels of 60 and above require immediate hospitalization. We mention this because if a level of 32 were considered alarming, we would have some evidence that the groups
were alarmingly different. Another approach of asking if this difference matters is
to examine the "spread" of scores. If scores within each group of children
have considerable spread, we could argue that 16 is "close" to 32. After
all, "closeness" is a relative term. The loose concept of
"spread" is most popularly measured by the standard-deviation.
The combined information of mean and standard-deviation tells us that the Exposed group had a mean level of
32, give-or-take 14, and the Control group a mean of 16 give-or-take 4.5. This is our first sign that something is seriously amiss with the Exposed children. Not only are the means different, but the standard deviations are substantially different.
It is difficult, and ill advised, to go much further without a picture.
A popular "rule of thumb" says that about 68% of the data lie within
1 SD of the mean, which means that most exposed children have lead levels between 18 and 46.
This means some children have alarmingly high lead levels, but this rule of thumb requires a symmetric distribution of
values. Thus we examine a picture of the distributions of the Exposed and Control
levels.
The Control group has a distribution that is quite symmetric. With
some thought, one should expect this. The Exposed group, on the other hand, looks
drastically different. Values are spread out, and at least two children are in need of immediate hospitalization. These two pictures say far more about the situation than a collection of summary
statistics, and provide strong evidence of....of what?
Certainly this is not proof that working at the battery factor caused this difference. But it is quite compelling evidence that a difference does exist. And requiring a significance test at this point is pedantic, since no reasonable person
would argue that there were no differences between these two groups.
But now we should ask what could account for these differences. We
might be quick to blame the factory, but students should be able to think of other causes. For
example, many of the Exposed kids might coincidentally live in an area that has a high level of lead in the environment (perhaps
in the paint used on their homes)? This area could be near the factory, which would
explain the discrepancy between the two groups. It might also be that the Exposed
children are older than the Control children -- for whatever reason -- and therefore have had more time for the lead to
accumulate.
As it turns out, the researchers also had these thoughts. (And
here is an example of some information somewhat unfairly hidden until a strategic moment.)
The researchers selected the Control group to "match" the Exposed group. For
each child in the Exposed group, another child of the same age living in the same neighborhood who did not have a parent working
around lead was chosen for the Control group. So the data actually consist of pairs of Exposed and Control children. How can
we exploit this fact when we examine the data?
Because the data are paired, it makes sense to examine the difference in lead levels for each Exposed child
and his or her pair. A dot-plot is revealing:
The vertical line highlights the fact that only four pairs had
negative values: these are pairs in which the Exposed child had less lead than the Control child.
One pair was the same, and the rest were all positive. In short, in only
four of 33 cases were the Exposed children better off than the Control. (Although we
should be careful of loose phrases like "better off" because a higher level of lead does not necessarily mean worse
health or greater risk.)
Because of the paired structure of the data, we have eliminated environment and age as explanations for the
differences between the groups. Another explanation is chance: were we to sample another 33 pairs, we might get a completely different outcome. This is a good point for a discussion about how to choose a statistical significance test for these data. Such tests come with assumptions, and by this point the students should be familiar enough
with the data set to question those assumptions. For example: can we think of these
pairs as randomly selected from some larger population? (Probably not. There is no reason to assume any random selection mechanism was used. The published report of the study also explains how difficult it was to find suitable
controls. It is also not clear exactly what the larger population should be in a
rigorous probabilistic setting. The set of all children of workers at this factory? At all battery factories in Oklahomah? At all
factories with lead in the environment?) One "probability model" might be
to think of these pairs as coin flips. If the observed outcomes are due solely to
chance, then it should be equally likely that an Exposed child will have more lead than a Control as the other way around. It is like flipping a coin: heads represents an
Exposed child having less lead than a Control. Tails
represents equal amounts or more. (Or you could throw the 0 observation into the
"heads" side if you wish.) Now if you flip a fair coin 33 times, what's the
probability that four or fewer will land heads? The probability is in fact
tremendously small (about 6 times 10 to the negative sixth for four or fewer heads, and 4 times 10 to the negative fifth for five
or fewer heads). Such small probabilities suggest that chance is not a good
explanation for the discrepancy. We must accept that something other than age,
environment, or chance is at play.
By this point we have eliminated causes, but have we "proven" that the factory is the cause? This is a good point for the association-is-not-causation discussion. But it is only fair to mention (more
strategically revealed information) that the study also reported a negative association between household cleanliness and lead levels in the children. In addition, Exposed children whose
parents showered and changed before leaving the battery factory had lead levels comparable to the Control group. Taken together this makes persuasive evidence that the factory is the cause, but does not "prove" that this
is so.
Lessons
Learned
I'd like to point out some themes of this case study:
-
Numerical summaries were incomplete; the story was better told through the dot-plots. The shape of the distribution was informative. The
fact that the shape of the distribution of the Exposed group was so different from that of the Control group was compelling
evidence of differences. This change of shape is missed by a standard numerical summary.
-
Scientific context guides the analysis. We calculate statistics
and make comparisons guided by our goal of examining a particular scientific issue.
-
The study design affects the analysis. Once we were told about
the paired structure of the data, we used a different analysis than we would had the data been unpaired.
Eight
Extra-Mathematical Themes
With the lead case study as a context, let's examine 8 themes or topics of Statistics that I consider to be
"extra mathematical."
1. Inference
The mathematical tools of probability allow us to quantify inference, but the basic act of inference --
generalizing about the whole based on a small part -- is more philosophical than mathematical. Particularly in this case study, in which our inference was causal -- did working at the
factory cause increased lead levels? -- mathematics can offer little support. Of
course we were not only interested in the 66 pairs of children at this factory. We
were concerned about any child whose parent's work exposes them to lead. How
can we generalize what we learned about these 66 children to any child? This is not a
mathematical issue.
2. Graphical Tools
Mathematics also relies on graphical tools, but differently. Both
Mathematics and Statistics want its students to learn to interpret graphs, but Statistics also wants its students to know how and when to use graphs. I relied on dot-plots above, but why not histograms?
Or box-plots? A data analyst needs to not only know which tools are available,
but also which will most likely be useful.
3. Calculation of Statistics
Statistics is not concerned with calculation. Yes, it is
important for students to produce the correct average of a list of numbers. But a
calculator or computer will do this for them. Statistics needs students to be able to
interpret these numbers in the context in which they were collected, and also needs students to know when to use which statistics
to help them answer questions. One could go a long way in analyzing the data from the lead case study without calculating any statistics.
What was more valuable was to know which statistics to calculate. (And, to be
frank, the mean and SD were quite possibly not the best choices, at least for the Exposed group.)
It was also important to know that the numerical value of the average was meaningless without knowing what a lead level of
32 meant medically, and without knowing how this compared to children who were not exposed to lead.
4. Science
Statistics is a tool of science, and as such is concerned with weighing evidence and not with proof. Science does not prove (although it may disprove). The
same can be said of Business, Engineering and the Social Sciences, all of which weigh evidence and make decisions, but only in
overly simplified situations are able to -- or even interested in -- "proving" in the mathematical sense.
5. Data Collection
The method by which the data were collected will affect the choices we make in our analysis. In the lead study, knowing that we did not have a random sample meant that we would have
to exercise extreme caution in applying a t-test. The data collection method also
affects the conclusions we might make. If the collection method was flawed, we might
not make any conclusions, but might still wish to provide a description of the data set. The collection method also imposes a
structure on the data, and our analysis should take this into account. Knowing that
the children were "paired" results in a different analysis. Mathematics says nothing about data collection methods, and in fact designing experiments that meet
mathematical assumptions of statistical tests is a craft.
6. Data Management
The first concern of a data analyst should be to check the quality of the data. This means more than looking into the data
collection method. It also means looking for influential observations, missing
values, and possible errors in entry. This is not such an issue in a data set as
small as the lead case study, but even there we might wonder the extent to which our conclusions were driven by one or two extreme
observations. (Not much, as it turns out.)
For large studies, such as most of the medical studies that are reported in our daily newspapers, data
management is an every-day concern. Data are collected on tens of thousands of
people, on perhaps hundreds (or thousands) of variables, at collections spread throughout the country or even the world. Organizing this data in a database, and retrieving the data required by a particular
researcher, requires considerable skill and training. Good and experienced database
managers these days can almost set their own salary. Even every-day consumers of
Statistics need to understand the extent to which the data themselves are constructed, stored, and retrieved.
As a side note, the analysis of what are called Massive Data Sets is an area of active research that includes
statisticians, mathematicians, computer scientists, and physical scientists.
7. Computer
The computer has a very special relationship to Statistics, and, if I might go out on a limb, it is a
relationship that mathematicians have a hard time understanding. The computer is much
more than a crutch that does tedious calculations very quickly. It is instead a
device that mediates between the data and the set of graphical and numerical tools at the statistician's disposal. Which software the statistician uses influences the course of the analysis. I would not go so far as to say that it affects the conclusions a statistician might make,
but I do think such an argument could perhaps be made. Certainly many journals insist
on knowing what software was used to analyze the data. An analogy can be made to
music. I have two recordings of Bach's Violin Partita No. 1. One is played by Jascha
Heifitz, the other is a piano transcription and played by the pianist Jorge
Bolet. Same
music, very different experience. Statistics software is an instrument in the same
sense a violin or piano is an instrument.
8. Consulting
By its very nature, Statistics is a collaborative activity. A
good analysis requires substantive knowledge, such as knowing that lead levels over 60 are extremely dangerous and that lead
accumulates in the blood over time. This means that Statisticians must collaborate. Mathematics is increasingly collaborative, but still it is possible for a mathematician to
be successful working alone. A statistician, though, almost always works in a
collaborative or consulting context, and needs to develop skills to assist in working with people from different
disciplines.
How To Train
Teachers?
Given these differences between Statistics and Mathematics, how, then, should high school teachers be
prepared? There are three words to keep in mind: practice, practice, and practice. Teachers
of statistics need to practice using computers to analyze real data, in realistic contexts. They
need practice choosing and applying tools. Ideally, I would like to think that at
least once they would be confronted with that confusing paralysis that happens when one faces a complex and large data set and
must choose how and where to begin. They should also have practice in collecting
data, or at least in dealing with data collected by different methods.
My own experience in training teachers comes through a Center which we founded within the UCLA Department of
Statistics: the Center for Teaching Statistics. The CTS disseminates information
about statistics teaching both to UCLA and to the local community. One of its
activities is to offer a course called "Data Analysis for High School Teachers." This
course is for current teachers of AP Statistics. Each week we work (using Fathom or
other statistical software such as ARC or Data Desk) on a different data set. The
course was designed after a meeting with a "focus group" of local AP Statistics teachers.
The consensus of that meeting was that while new Statistics teachers had the background necessary to conquer the
mathematical content of Statistics, many were unprepared to deal with the applications side.
Many teachers, for example, felt uninformed in assigning or evaluating student projects.
This course, by providing participants with practice working with real data, helps them develop understanding of how actual
statisticians apply tools.
I have also been involved with a program run by the UCLA
Mathematics Department called the UCLA Math Content Program for Teachers. This
program is designed for current teachers who are "under-prepared" to teach mathematics. It consists of a two-year series of courses than focus on 4th to 9th grade mathematics. Dealing with Data, an introductory statistics
course, is now the most popular of their second-year offerings. The course teaches
the basic tools of statistics and is activity-oriented, involving participants in
examples pulled from real-life. This course makes frequent use of simulations, both
as an aid to understanding probability concepts, and also as a problem solving tool. An
important feature of the course is the end-of-term data collection and analysis project. The
Math Content Program is directed by Dr. Shelley Kriegler, and more information can be found at
www.math.ucla.edu/teachers.
In conclusion, my belief is that training mathematics teachers to teach statistics requires a substantial
effort. This effort goes beyond the traditional mathematical statistics course most
math majors take (if they take any statistics class at all), and requires at least one course -- perhaps called "data
analysis"? -- that teaches this extra-mathematical content.
REFERENCES to Part I
1.
Burrill, G., Kranendonk, H., Landwehr, J., Scheaffer, R., Witmer, J., etc., Data-Driven Mathematics (11 modules),
Columbus, OH, Pearson Learning, 1998. www.pearsonlearning.com
2.
Landwehr, J., Scheaffer, R., Watkins, A., etc., The Quantitative Literacy Series (5 books), Columbus. OH,
Pearson Learning, 1995. www.pearsonlearning.com
3.
Franklin, Christine, “Are Our Teachers Prepared to Provide Instruction in Statistics at the K-12 Level?”, NCTM Mathematics Education Dialogues, Vol.37, Issue 3, October 2000.
4.
Franklin, Christine, “Exploring Data: How Students Answer Questions on the AP Statistics Exam and What are the Desired
Responses”, American Statistical Association’s 2000 Proceedings of the Section on Statistics Education, pp.192-197.
REFERENCES to
Part II
1.
Morton, D., (1982), "Lead absorption in children of employees in a lead-related industry," American Journal of Epidemiology, Vol. 155, pp. 549-555.
2.
Trumbo, Bruce, (2001), Learning Statistics with Real Data, Duxbury Press
3.
Gould, Robert and Bruce Trumbo, "Lead Absorption Case Study", appearing in Cyberstats, Cybergnostics Inc., to be published. www.cybergnostics.com
Software
Fathom: www.keycollege.com/Pages/ProdFathom.html
DataDesk: www.datadesk.com
ARC: www.stat.umn.edu/arc
JMP: www.jmpdiscovery.com
Stata: www.stata.com
SAS: www.sas.com
My opinions on choosing software can be found at www.stat.ucla.edu/~rgould/met
Related Links
www.stat.ucla.edu/~rgould/met outline from presentation, and software advice
www.stat.ucla.edu/cts
the UCLA Center for Teaching Statistics
www.math.ucla.edu/teachers
the UCLA Math Content Program for Teachers
Appendix: Activity
Chris Franklin
A Case of Possible Discrimination
Statisticians are often asked to look at data from situations where an individual or individuals believe that
discrimination has taken place. A well-known study of possible discrimination was reported in the Journal of Applied Psychology. The scenario of this study is given below.
Scenario
In 1972, 48 male bank supervisors were each given the same personnel file and asked to judge whether the
person should be promoted to a branch manager job that was described as “routine’ or whether the person’s file should be
held and other applicants interviewed. The files were identical except that half of the supervisors had files showing the person
was male while the other half had files showing the person was female. Of the 48 files reviewed, 35 were promoted.
(B.Rosen and T.
Jerdee (1974), “Influence of sex role stereotypes on personnel decisions,” J. Applied
Psychology, 59:9-14.)
Preliminary Questions
1.
Suppose there was no discrimination involved in the promotions. Enter the
expected numbers of males promoted and females promoted for this case in Table1.
Table
4. Actual discrimination study
5. What percentage of males was recommended
for promotion? What percentage of females was recommended for promotion?
6.
Without exploring the data any further, do you think there was discrimination by the bank supervisors against the females?
How certain are you?
7.
Could the smaller number of recommended female applicants for promotion be attributed to chance? What is your sense of how likely the smaller number of recommended females could have
occurred by chance?
8.
Suppose that the bank supervisors looked at files of actual female and male applicants. Assume that all of the applicants
were identical with regard to their qualifications and use the same results as before (21 males and14 females). If a lawyer
retained by the female applicants hired you as a statistical consultant, how would you consider obtaining evidence to make a
decision about whether the observed results were due to chance variation or if the observed results were due to discrimination
against the women?
Statisticians would formalize the overriding question by giving two
statements, a null hypothesis which represents no discrimination so that any departure
from Table 1 is due
solely to a chance process, and an alternative hypothesis which represents
discrimination against the women.
9.
What initial thoughts do you have about the manner in which the experiment
(study) was conducted? What would need to be assumed about the study in order to infer that gender is the cause of the
apparent differences?
[PSSM quote: Students should
formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.]
Simulation of the Discrimination Case
Using a deck of cards, let 24 black cards represent the males, and 24 red cards represent the females (remove
2 red cards and 2 black cards from the deck). This will simulate the 48 folders, half
of which were labeled male and the other half female.
1:
Shuffle the 48 cards at least 6 or 7 times to insure that the cards counted out are from a random process.
2:
Count out the top 35 cards. These cards represent the applicants recommended for promotion to bank manager. The simulation
could be conducted more efficiently by dealing out 13 “not promoted” cards, which would be equivalent to dealing out 35 “promoted”
cards.
3: Out of the 35 cards,
count the number of black cards (representing the males).
4:
On the number line provided, Figure 1, create a dot plot by placing a blackened circle or X above the number of black cards
counted. The range of values for possible black cards is 11 to 24.
5: Repeat steps 1 – 4
nineteen more times for a total of 20 simulations.
FIGURE 1
DOT PLOT TO BE USED TO GRAPH THE 20 SIMULATED RESULTS
____________________________________________________________________
11
12 13 14 15
16 17
18 19 20 21 22 23
24
Number of Men Promoted
[PSSM quote: Students should use simulations to explore the variability of sample statistics from
a known population and to construct sampling distributions.]
A sampling distribution is the distribution of possible values of a statistic for repeated
samples of the same size from a population. For the scenario under consideration, the number of black cards (number of males
promoted) from each of the simulations is the statistic.
6:
Using the results (the counts) plotted on the number line, estimate the chances that 21 or more black cards (males) out of
35 will be selected if the selection process is random; that is, if there is no discrimination against the women in the selection
process.
The probability of observing 21 or more black cards if the selection
process is due to randomness or chance variation is called the p-value.
7:
Look at the dot plot and comment on the shape, center, and variability of distribution of the counts by answering the
following.
(a)
Is the distribution somewhat symmetric, pulled (skewed) to the right, or pulled to the left?
(b)
Do you observe any unusual observation(s)?
(c)
Where on the dot plot is the lower 50% of the observations?
(d)
Estimate the mean of the distribution representing the number of black cards obtained out of the 20 simulations.
(e)
What values occurred most often?
(f)
Finally, using the dot plot, comment on the spread of the data.
[PSSM quote:
Students should select and use appropriate statistical methods to analyze data for univariate measurement data, be able to display
the distribution, describe its shape, and select and calculate summary statistics.]
8. Is the behavior of
this distribution what you might expect? Why or why not?
9.
Think about the question of possible discrimination against the women. Based on the exploration just made, does there
appear to be evidence to support the claim that selecting 21 males (black cards) for
promotion out of 35 candidates was not due to chance variation?; i.e., how does your data compare with that of the original study?
10.
How do your results compare with the results of your classmates?
Using Probability Theory
In attempting to answer the question, “ Is the difference between the number of observed promoted males and
the number of expected male due to chance variation or is this difference a real difference?”, simulation provided an estimated
probability of 21 or more males being promoted out of 35 promotions. This probability can also be found by the application of
probability theory. The stated question of interest leads to a situation that results in “ success-failure” outcomes. A group
of N population objects is classified into two subgroups, where one group of size k
is the success group and the other group of size N-k is the failure group. The number of
objects selected from the population is denoted as n.
Since a male being promoted was defined as the variable of interest, a male being promoted is a success; a female being
promoted is a failure. For this discrimination activity, N = 48 applicants, k = 24 males, and N-k
= 24 females. Of the 48 applicants, n=35 were selected for promotion. A random variable X is defined that represents the number of successes out of n; that is, the number of males selected out of the 35 promoted. The probability of a
success occurring on any given promotion is k/N = 24/48 = 0.50. The expected value for
the number of males to be promoted out of the 35 would equal 35(0.50) = 17.5.
[The
expected value of a random variable X is another name given to the mean of the random variable X.]
[PSSM quote:
All students should be able to compute and interpret that expected value of random variables in simple cases.]
The selection of 35 of the 48 applicants for promotion is viewed as sampling without replacement; that is,
once an applicant is selected for promotion, the applicant is not returned to the population of applicants, which does not provide
the opportunity for another evaluation and promotion. Mathematically, the probability of selecting 21 or more males for promotion
out of the 35 promoted can be found as follows. Let N = k + (N-k), n = number of objects selected from the population of size N, and X = number of successes out of n.
The desired probability is P(X
21) = P(X=21)+P(X=22)+P(X=23)+P(X=24).
To find the probability that exactly X of the males were promoted, we must first find the number of ways to select x
males from k = 24 males in the population and the number of ways to select (n-x) females from (N-k) females in the
population. This can be found by evaluating the combinations
and
. By the multiplication principle, the product
equals the number of ways of promoting x
males and n-x females. The combination
gives the total number of ways of promoting n applicants out of N. Thus, P(X=x) =
/
.
[The random variable X follows a hypergeometric distribution.]
[PSSM quote:
Students should be able to represent and analyze mathematical situations using algebraic symbols.]
Evaluating for
P(X=21) =
OWS/TEMP/msoclip1/01/clip_image036.pcz" o:title=""/>
/
= [24!/(21! 3!)] [24!/(14! 10!) / 48!(35!13!) = 0.021
P(X=22) =
/
= 0.004
P(X=23) =
= 0.000
P(X=24) =
=
= 0.000.
Therefore, P(X
21) = 0.021 + 0.004 + 0.000 + 0.000 = 0.025.
This mathematical probability is a long-term probability. It is the percentage of time that an outcome would
be expected to occur in the long run if a simulation or experiment was replicated many times. Observe how close the mathematical
probability of 0.025 is to the simulated probability of 0.028 from the class data sets considered in Sample Distribution 4 and
Sample Distributions 5.
[ Refer to a
high school mathematics book that covers combinations for additional details on how to evaluate a combination and evaluating
factorials.]
