a student attempts a multiple choice exam

8/14/2019 A Student Attempts a Multiple Choice Exam

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Example

A student attempts a multiple choice exam

(options A to F for each question), but

having done no work, selects his answers toeach question by rolling a fair die (A = 1, B =

2, etc.).

If the exam contains 100 questions, what is

the probability of obtaining a mark below 20?


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Simulation

Now, let us simulate a large number of

realisations of students using this random

method of answering multiple choicequestions. We still require the same

Binomial distribution with n=100 and a=

This can be done on R using the command

rbinom.

1

6


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For example, lets simulate 1000 students.

> xsim=rbinom(1000,100,1/6)

> xsim [1] 18 22 9 17 18 20 21 16 8 18 11 16 16 13 16 14 25 15 16 17

[21] 13 25 11 24 17 16 13 21 10 17 18 10 17 18 19 17 19 15 13 12

[41] 15 11 21 23 19 14 19 25 23 19 20 17 17 15 16 14 13 16 17 14

[61] 24 21 19 8 18 20 22 16 15 20 19 17 13 15 13 21 22 12 12 12

[81] 11 14 11 12 16 16 17 21 17 16 17 14 9 17 16 17 12 20 16 17

[101] 18 13 15 16 12 15 17 16 17 26 18 14 21 15 10 23 12 16 16 12

[121] 17 18 22 17 18 14 19 22 13 17 21 15 21 16 17 16 16 28 16 17

[141] 18 19 16 11 14 18 16 18 18 14 20 13 19 19 22 22 13 17 19 17

[161] 18 20 11 22 19 25 15 15 17 18 5 15 14 13 18 15 17 15 20 17

[181] 16 14 23 17 16 10 12 16 21 30 16 13 22 14 15 16 17 14 16 18[201] 14 20 16 19 25 14 15 24 22 19 15 17 22 10 20 13 10 15 14 22

[221] 17 12 16 19 20 17 15 21 14 13 21 11 19 9 21 22 16 13 13 12

[241] 14 13 18 8 14 18 10 16 10 12 21 18 15 17 16 8 19 17 11 18

[261] 23 17 20 16 12 20 11 16 22 17 16 13 22 20 15 15 20 17 22 14

[281] 18 23 18 20 20 16 19 16 15 19 18 17 14 22 15 24 17 15 17 22


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[301] 18 22 10 19 24 21 16 14 11 14 20 15 21 11 17 16 20 19 13 14

[321] 17 17 19 15 17 13 18 23 16 12 25 13 13 21 19 16 20 27 19 18

[341] 18 24 15 23 13 13 14 15 23 13 19 15 11 19 17 12 15 15 17 14

[361] 18 20 17 13 16 14 13 20 18 15 18 16 17 20 14 19 21 12 13 17[381] 22 17 19 16 14 18 16 18 12 16 13 15 16 9 15 16 18 22 14 16

[401] 14 17 12 16 21 16 21 13 14 19 18 18 16 19 17 17 17 13 17 11

[421] 16 16 13 10 26 12 20 17 11 19 18 12 15 14 14 20 15 15 15 11

[441] 18 23 20 23 13 12 18 22 12 16 13 21 22 14 18 21 17 12 19 16

[461] 17 18 15 22 22 20 15 16 13 12 19 22 16 20 19 19 16 8 15 12[481] 29 26 19 16 20 15 11 22 15 20 21 14 16 13 17 15 10 13 17 12

[501] 18 20 17 14 13 19 23 11 27 19 17 16 17 20 21 15 20 20 21 19

[521] 21 16 13 21 16 19 13 9 10 20 12 18 14 13 18 19 22 19 21 18

[541] 6 17 17 19 19 22 23 18 13 12 17 16 21 16 18 21 19 13 22 19

[561] 20 17 18 15 17 15 15 10 18 13 23 17 14 23 22 10 18 11 11 18[581] 16 17 14 13 9 12 14 14 21 23 24 19 12 15 17 18 11 14 19 19

[601] 19 16 17 13 13 15 17 18 17 13 9 19 18 22 17 13 14 22 13 23

[621] 23 19 19 16 24 14 17 18 17 13 16 12 7 15 17 16 18 22 19 15

[641] 16 18 18 13 20 18 12 6 15 11 16 19 12 13 11 17 11 15 11 19


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[661] 17 16 16 21 12 18 20 19 16 14 18 17 16 14 11 17 17 16 17 17

[681] 17 18 16 18 12 18 18 20 19 13 12 16 14 13 13 6 15 12 19 14

[701] 20 17 16 14 21 19 15 26 17 20 12 24 13 11 19 21 18 13 9 16

[721] 9 16 17 16 15 12 11 21 21 13 19 13 13 16 11 17 15 19 22 19[741] 11 13 14 16 20 15 16 12 18 14 12 14 21 12 23 21 19 10 24 17

[761] 17 19 19 15 18 12 14 14 14 20 12 20 12 21 19 20 21 20 17 18

[781] 15 12 16 23 16 16 19 15 12 14 21 25 12 19 20 22 17 16 21 20

[801] 23 24 17 20 17 19 14 22 20 25 10 12 15 16 7 14 14 18 22 10

[821] 15 22 23 18 12 10 14 18 15 15 18 10 21 11 20 15 20 10 13 16[841] 16 17 22 19 19 16 8 20 17 13 21 16 25 16 13 17 14 17 19 21

[861] 17 19 14 22 20 18 14 19 17 23 20 18 14 11 16 18 26 24 24 18

[881] 21 16 23 20 14 16 15 13 14 11 12 13 14 16 18 17 16 17 13 20

[901] 22 8 17 17 16 16 14 22 17 18 18 21 15 11 20 21 18 15 19 21

[921] 16 22 14 12 16 20 16 21 11 13 19 14 23 12 12 17 14 15 26 17[941] 18 14 21 17 14 24 21 12 21 13 20 22 11 20 10 16 16 15 19 13

[961] 16 15 16 17 9 14 11 12 19 17 16 15 21 14 15 14 15 17 15 16

[981] 19 11 15 17 17 17 11 18 21 14 15 17 18 16 11 22 19 16 14 15


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It makes sense now to look at properties of

these 1000 simulations which have been

placed in the vector xsim.

> mean(xsim)

[1] 16.624

> median(xsim)

[1] 17

> sd(xsim)

[1] 3.778479> var(xsim)

[1] 14.2769

>


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Now compare the actual values from the

simulations, with the theoretical values fromthe probability distribution.

SIMULATION THEORETICAL

MEAN 16.624 16.66667

VARIANCE 14.2769 13.88889


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A full summary of the results of the simulation

is given with:

> table(xsim)

xsim

5 6 7 8 9 10 11 12 13 14 15 16 171 3 2 7 10 21 40 57 72 80 82 118 118

18 19 20 21 22 23 24 25 26 27 28 29 30

85 83 61 55 46 25 14 9 6 2 1 1 1>


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A Histogram can also be plotted of this:

> hist(xsim)


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Notice that a BARPLOT of xsim does

NOT produce a useful graph!

> barplot(xsim)


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A barplot of the TABLE of xsim does

work,though.

> barplot(table(xsim))


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Poisson Distribution

The Poisson distribution is used to model thenumber of events occurring within a given time

interval. The formula for the Poisson

probability density (mass) function is

is the shape parameter which indicates the

average number of events in the given time

interval.

( )

!

xe

p x

x

=


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Some events are rather rare - they don't

happen that often. For instance, car

accidents are the exception rather than therule. Still, over a period of time, we can say

something about the nature of rare events.

An example is the improvement of traffic

safety, where the government wants to know

whether seat belts reduce the number ofdeath in car accidents. Here, the Poisson

distribution can be a useful tool to answer

questions about benefits of seat belt use.


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Other phenomena that often follow a Poisson

distribution are death of infants, the number of

misprints in a book, the number of customersarriving, and the number of activations of a

Geiger counter.

The distribution was derived bythe French mathematician

Simon Poisson in 1837, and

the first application was thedescription of the number of

deaths by horse kicking in the

Prussian army.


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Example

Arrivals at a bus-stop follow a

Poisson distribution with an average

of 4.5 every quarter of an hour.

Obtain a barplot of the distribution

(assume a maximum of 20 arrivals in

a quarter of an hour) and calculatethe probability of fewer than 3 arrivals

in a quarter of an hour.
http://images.google.co.uk/imgres?imgurl=http://www.nataliedee.com/painting-busstop.jpg&imgrefurl=http://www.nataliedee.com/newpaintings.php&h=687&w=364&sz=31&tbnid=dac4lfht2FwJ:&tbnh=136&tbnw=72&start=2&prev=/images%3Fq%3Dbusstop%26hl%3Den%26lr%3D


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The probabilities of 0 up to 2 arrivals

can be calculated directly from theformula

( ) !

xe

p x x

=

4.5 0

4.5(0)0!

ep

=

with =4.5

So p(0) = 0.01111


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Similarly p(1)=0.04999 and p(2)=0.11248

So the probability of fewer than 3 arrivals

is 0.01111+ 0.04999 + 0.11248 =0.17358


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R Code

As with the Binomial distribution, the

codes

dpois and

ppoiswill do the calculations for you.


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> x=dpois(0:20,4.5)

> x[1] 1.110900e-02 4.999048e-02 1.124786e-01 1.687179e-01 1.898076e-01

[6] 1.708269e-01 1.281201e-01 8.236295e-02 4.632916e-02 2.316458e-02

[11] 1.042406e-02 4.264389e-03 1.599146e-03 5.535504e-04 1.779269e-04

[16] 5.337808e-05 1.501258e-05 3.973919e-06 9.934798e-07 2.352979e-07[21] 5.294202e-08

>


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> barplot(x,names=0:20)


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Now check that ppois gives the same answer

(ppois is a cumulative distribution).

> ppois(2,4.5)[1] 0.1735781

>


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Consider a collection of graphs fordifferent values of


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=3


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=4


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=5


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=6


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=10


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In the last case, the probability of 20arrivals is no longer negligible, so

values up to, say, 30 would have to be

considered.


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Properties of Poisson

The mean and variance are both equal to

.

The sum of independent Poisson variables

is a further Poisson variable with mean

equal to the sum of the individual means.

As well as cropping up in the situations

already mentioned, the Poisson distribution

provides an approximation for the Binomial

distribution.


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Approximation:

If n is large and p is small, then the

Binomial distribution with parameters n andp, ( B(n;p) ), is well approximated by the

Poisson distribution with parameter np, i.e.

by the Poisson distribution with the samemean


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Example

Binomial situation, n= 100, p=0.075

Calculate the probability of fewer than10 successes.


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> pbinom(9,100,0.075)[1] 0.7832687

>

This would have been very tricky with

manual calculation as the factorials

are very large and the probabilitiesvery small


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The Poisson approximation to the

Binomial states that will be equal

to np, i.e. 100 x 0.075

so =7.5

> ppois(9,7.5)[1] 0.7764076

>

So it is correct to 2 decimal places.

Manually, this would have been much

simpler to do than the Binomial.

a student attempts a multiple choice exam

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