DOCUMENT RESUME

ED_ 079'985 EM 011 3564 ,

AUTHOR . Tanis, Elliot A._TITLE 'A computer Laboratory for Mathematical Probability

and Statistict.. ..

.

INSTITUTION Hope Coll., Holland, MiCh._ .

SPANS AGENCY National Science Foundation, Washington-, D.C.,PUB DATE Jdn 73' .

NOTE 11p.; Paper presented at the Conference on Computersin the Undergraduate Curricula-(Claremont,:California, June 1e-20, 1973)

EDRS PRICE MFz$0.65 HC-$3-.29DESCRIPTORS College Instruction; *College Mathematics; Computer

Assisted_InstrUction; *Computer Based Laboratories;Higher Education; Mathematics- Instruction; -

*Probability Theory; Program Descriptiops; Speeches;- *Statistics; StudentAttitudes --

IDENTIFIERS Hbpe College; IBM 1130-Computer. .

ABSTRACTA ptoject-conc#inediath-the.modifiaation-of a

=Mathematical pkobabiiit-k and- statittidscOUtse_ is described._The-CO--Ai-tie was redesigned -to include- a.Week4-00:_hoUr-libpratory session411;Which-students-use-an IBM 1110-cOniputer tb_iteiify:e0pikically-the-e0retical pOofs,rearned: in tbLe_clitSroom.--_Thia project report

i=the eutirontines-fntniihed-to-the=ttOdents-and-the-topicsHofeach of the 15 lab sessions. In additiohl:deediipti:opt of three-1Abbratory-exerOises are given bymay_of.iiluetkationi fOlikpmed by a=summary -of student,teactiOnt and fadOity attitudes toward the use-of;the cothpUtet fot ldboratOry siori..(LB)

FROM BEST*AVAILABtE COPY'

rtA Co".fer-evIce b% Co 4,1944 i'ersPA) /Wed i" Proceedi'njr 04:

#01.e .TA e fey- 49, / 734. L: /eAremift Ct"A COMPUTER LABORATORY FOR MATHEMATICAL PROBABILITY AND STATISTICS*

Elliot A. TanisHope College

Holland; Michigan 49423(616) 392-:5111-

1 . . .For a number of years the junior- senior level mathematical probability and statistics'course at Hope College- was a two semester course - meeting for lecture and discussion threeperiods per lure,O., This -project*-concerns,itself with a modification of.the.courseso.that alaboratory which meets two hours each week is an integral paft °fit. one hour of Credit._ isgiven-each semester for this, laboratory which makes*extensiie use of our IBM 1130 computer.'

*.

.

.___, .

The deve16-Pient of the laboratory materials is taking place over a 27 month periO4, 'June, 1971, through August, 1973. Atthe 1972 Conference on Computers in the UndergraduateCarr icdfathis-projedt_was described and a listing of the laboratories for the first semesterwas giiev[ i]. We are now able to list the laboratotiee for the -Second semester, describesome of them, and give a better overall evaluation.ofthe laboratory. . *.

,

Laboratotx DesCriplion'

In' the laboratory-the students illuttrate empirically many of the theoretical conceptswhich are- proved in the classtoom. By writing computer ,programs they gain a better -anddeeper understanding of the theory.- In addition they become aware of the capabilities of-thecomputer for= analyzing data..

r We generally -spend the first hour of the laboratorydiscussing. the assignment of theprevious Week and introducing the new assignment. The second hour of the laboratory is usedto design and.writi the program. The format.of the laboratory, would probably be'different-ifthe e-students had access to -terminals.

Several subroutines are furnished for the students. These includp:

(1) a random .numbet generator,

(2) plotting .subroutines for (a) empirical- and theoretical distribution functions,(b) relative frequency histograms and ,probability density. functions, and (a)scatter diagrams with the option of adding the least squares regression line-,

A3) programs which give values of a number of distribution functions, and(4) subroutines which select random samples from the standard distributions.

Phe following listing gives the topics covered.in each of the 15 laboratories during thesecond semester. In section 3 we shall describe three experiments.

U S DEPARTMENT OF HEALTH,EDUCATION SWELFARFNATIONAL INSTITUTE OF

EDUCATIONTHIS 'DOCUMENT HAS SEEN REPRODUCED' EXACTLY AS NecEtveo ;NomTHE PERSON OR ORGANIZATION ORIGINA10.0 IT POINTS OF VIEW OR OPINIONSSTATED DO NOT NECESSARILY REPRESENT OFFICIALNATIONAL INSTITUTE OFEDUCATION POSITION OR POLICY

=

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Laboratory

1

2

3

Topics

Transformations of Random Variables, Cauchy Distribution, N(0,1)Distribution Using Box-Muller Method.

chi-square and Beta Distributions.

?unctions. of Independent and Identically Distributed Normal mendo,Variables.

4 . The t.and r Distributions.

5 Convergence in Probability; Convergence in Distribution.

6 Central Limit Theorem.

7 Normal Approximations for the Binomial Distribution.

8 Estimation.

9 Confidence intervals for Means and Variances;

Confidence Intervals.f or.Proportions.

'11 Tests of SimpleBypotheses.

-12 Thi Power Ptnction.

13 Analysis of variance.

14 A Test of the Equality of Multinomial Didtributions.

15 order Statistics.

LabordtorxExamies

We shall give a brief description of.. parts of three latoratories.

Laboratory 4 - The t and F Distributions: A t random variable with r = 8 degrees offreedam is'defined,by

Z.T =

497Nwhere Z has a normal distribution with seaniu=0 and variance c72 =1 thereafter N(0,1)1, U has achi - square distribution ith r=8 degrees of freedom [hereafter 1L2(8)),' Z and U are

independei. -

fhe students are able to simulate samples from both the normal and chi-square

distributions. Thui they are able to simulate n=300 observations Cf r and obtain a histogramsimilar to that shown in Figure 1. -This is' a relative.freguencyhistOgram iriti the;mobability density function for the t distributionNit* r=8 degrees of freedom

-

superiiposed.-

The importance of the t distribution is its use in normal sampling. To illustrate this,

let X1' 2

X2, . -X be a random sample front a distribution which is N(30,16). Then

-30

(x,i-x)( 8) (8)

has a t distribution with r=8 degrees of freedom. A random sample-of size n =300 observationsof T, generated using this ielaticnship, yielded the hiStOg.rim in Figure 2.a

Students are able to raise arid partially answer questions concerning the hypotheses of

certain theorem's. ?or-example, what if theyando sample in the above exercise does, not come

from a normal distribution. Does T defined above still have a t distribution?

3r suppose the sample does coke from a normal distribution. What effect does roundingoff the observations have.on the distribution of T? This question was investigated by one of

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Laboritor 6 - The Central Limit Theorem: (This particular example was sotivated by thework of two Hope College students (3,4].)

Tits _experiment 'illustrates the Central_Lieit Theorem and gives students some feelingfor lebiding when n is "sufficiently largee tor using the normal approximation. Thisparticular example also helps to emphasize that the Central Limit Theorem is applicable tocontinuous,-discrete, and mixed distributions.

:onsider the following game. 'Flip a coin. rE the outcome is heads, *in $2: if theoutcome is tails, spin a balanced spinner which his outcomes between 0 and 1 with the payoff'equal to the outcome on the spinner. Let X denote&the payoff for this game. 'Then X has a-mixed eistribetion,p=5/44 az=29/48. Let XI, X2, . . . , Xn denote the outcomes on nindependent plays of this game. Then . AW

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converges in distribution to the- random variable V which is le(0,1) .

-In the laboratory we would ilrnstrate that TMn has a-distribution which is approximatelyS(00) vhen n is sufficiently large. In this paper we shall use two values of n,- n=1 andn=13: This game vas simulated'200 times yielding 200 observations of VI. The histogram of

--these datA is plotted in Figure 3,:eith the probability density function -for the normallittribution,p=0,. 442=1 superimposed. An observation of.the random variable 413 is Simulatedusing- 13 independent plays of the game. Thus 200 observations of Um are- based _upon 2600plays of the game. The hittogram of 200 obiervailons of V13 is given in Figure 4 with them(0,1) probability density function superimposed. The empirical -distribition- function for

the first 25 observations of 413-is plotted along with fhe V(0,1) iistribetion function inFigure 5. .

Let ill,- X2, . . , Xn denote a random sample of size n from a distribution which hasseenp and variance ez. Let

V =n' 0,6117

A "rule of thumb" given in many -textbooks is that Nn -has a distribution which isapproximatelre(0,1) when n is greater than 25 or 30. Using the techniques learned in theabove- and similar exercises, a student.ii able to test thisA0rule of thumb', empirically forserveral-distributions.

Laboratory 12 - The Power Function: This laboratory illustrates the power-function boththeoretically and empirically. We shall describe the particular exercise which uses a test

oh the variance..;

Get X have a normal distribution with mean)u and variance az. we shall test the nullhypothesis-110:a2=47.5 against the alternative hypothesis-111:0,<47.5.

If the test is based upon a sample of size n=10, a critical region for a significancelevel of

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FIGURE

6

re Figure 6 the power function is graphed along with.the relative frequencies of

rejections for the 10 values ofcr2.

With the techni es learned in .this example, a student is.able to compare two power

functions for the Same est which differ only in the size of the critical region. A student

is° also able to define a power function empirically when he is perhaps not able to define it

theoretically. -

Student Reaction

This laboratory has been taken by two classes of junior and senior mathematics and

science majors. The general reaction of most of the students -'to the laboratory has been-

favorable. Of the -25.students in this year's class who responded'to the question "'Did the

laboratory help you understand more clearly the theoretical concepts?, 19 said yes, 4 said

no, and 3 said somewhat...

3ne student thought the laboratory was very helpful in highlighting the material in

class. Another said it took a lot of tine but was interesting and helpful. 'Still- another

student -felt that many of the theoretical principles of.dtatistics were reinforced by the lab

moil. A few students suggested that the use of more data which had been collected-in science

laboratories would have been.interesting. For one student the laboratory was not beneficial

bidause she generallybelieved the theoretical concepts without the empirical printout.

I believe that in general the reaction of a student to the liboratorf depends largely

upon his-like or dislike of the computer.

Faculty Evaluation

August 14-18, 1972, we held a conference at-lope_College for 51 college teachers of

statistics-from the midwest and eat.. During this conferene-ire described our use of the

computer in Our statistics -curriculum. From this conference we received both written and

verbal comments, some of which I would like to summarize.

It is' not always =possible to add a laboratory -because of insufficient staff time Or

ir,adeauats scaputsi facilities. 'In such cases it is perbeps possible for the students to

writs only a few comvuter-prbgrams while the_instructor illustrates most of the ideas with -

printouts or transparencies. However mathematics majors might be less intrigued with just

seeing others' results without -actually being able to perform the exercised on their own.

Also students seem tc gain intuition when they use the computer.

A valuable by-product of the statistics laboratory is the opportunity for students-to do

some research as undergraduates. Their mathematical- maturity and background may not be

sufficient to 'prove certain - results _theoretically but they are able to obtain answers

empirically.

Some of our conference participants are-using part of our materials during the 1972-73

academic year. We hope to receive summaries of their experiences by the end of this academic

pear.

Summail

A computer laboratory in statistics provides new and interesting_challenges to both

- student-and teacher. A lot of extra time is required. However if the exercises are

meaningful and not just busywork, the time-is well spent,

Y--

1.

REFERENCES

Elliot A. Tanis, Theory of. probabil.t.ty weld statistics illustrated by the computer.Proceedings of the 1972 Conference on-Computers in undergraduate Curricula, 1972, 513--

.520.'

2. Joyce A. Newell, Effects of violation of kyootheses for the t distribution. UnpublishedSenior Hokors Project, Hope College, 1970.

Deanna Gross Snyder, How large is large enough? A study of the ramifications of the:entral Limit Theorem through simulation. techniques. ,Unpublished Senior .Honors Project,Hope College; 1968.

U. Patricia Lang Young, kpplicaticin of the Central limit Theorem to a mixed distribution.Unpublished Independent Study Project, Hope College, 1969.

* This-project is being supported by the'lational Sbience Foundation through Grant G.- 28786.