11mab_-_2010_-_assessment_2.2_-_empt[1]

8/8/2019 11MAB_-_2010_-_Assessment_2.2_-_EMPT[1]

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YEAR 11 MATHEMATICS B

Name: Teacher: SPY JJH KAF CLJ

Assessment Instrument: 2.2 EMPT

Subject Matter: Applied Statistical AnalysisDate Set: Friday 17

thSeptember 2010

Date Due: Friday 22nd

October 2010

Duration: 3 weeks (1 class lesson allocated for student work)

Instructions:

Attempt all questions showing full working and reasoning Consider the Criteria Sheet when addressing objectives. Exploring ALL solutions is essential when making decisions on models A Graphics Calculator, Computer Graphing software may be used where appropriateYour answers will be assessed according to the following criteria:

(Modelling and Problem Solving and Communication and Justification are further detailed on the reverse)

Knowledge andProcedures

Modelling and ProblemSolving

Communicationand Justification

QuestionNumber

Recall,access,selectionofmathematicalde

finitions,

rulesandproceduresinroutineandnon-routinesimple

tasksthroughtoroutinecomplextasks

Applicationofmathematicaldefinitions,rule

sand

proceduresinroutineandnon-routinesimp

letasks,

throughtoroutinecomplextasks,

Numericalcalculations,spatialsenseanda

lgebraic

facilityinroutineandnon-routinesimplet

asks

throughtoroutinecomplextasks,

inlife-re

latedand

Appropriateselectionandaccurateuseoftechnology

Useofproblem-solvingstrategiestointerpret,

clarifyandanalyseproblemstodevelo

p

responsesfromroutinesimpletasksthroughto

Identificationofassumptionsandtheirasso

ciated

effects,parametersand/orvariables

Useofdatatosynthesisemathematicalmo

delsand

generationofdatafromm

athematicalmodelsin

simplethroughtocomplexsituations

Investigationandevaluationofthevalidity

of

mathematicalargumentsincludingtheanalysisof

resultsinthecontextofproblems;thestrengthsand

Appropriateinterpretationanduseofmathematical

terminology,symbolsandconventionsfrom

simple

throughtocomplex

Organisationandpresentationofinformatio

nina

varietyofrepresentations

Analysisandtranslationofinformationfrom

one

representationtoanotherinlife-relatedandabstract

situations

Useofmathematicalreasoningtodevelopcoherent,

conciseandlogicalsequenceswithinaresp

onsefrom

simplethroughtocomplex

Coherent,conciseandlogicaljustificationo

f

procedures,

decisionsthereasonablenesso

fresults

1, 2, 3, 5, 8

4, 6, 7, 9

/ 33* * ** ***

Standard


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St Pauls School Mathematics Department

Modelling and Problem Solving Criteria

AThe problem is fully solved. The response shows evidence of interpretation, analysis, identification of assumption, selections, use of synthesis of

strategies and procedures while showing initiative to solve the problem. Minor computational errors may be evident.

BThe problem has not been solved; however, substantial progress towards a rational solution and the response shows evidence of appropriate

interpretation, analysis, identification of assumptions and selection of strategies and procedures.

C The student has made a reasonable interpretation of the problem and has selected strategies and/or procedures appropriate to the problem.

D The student has followed basic strategies and/or procedures appropriate to the problem.

E The student is unable to begin the problem or hands in work that is meaningless. Or the student has provided only an answer.

CAJ DESCRIPTORS 1 2 3 4 * 5 6 ** 7 * 8 9 *** TOTAL

Appropriate interpretation and use of

Terminology, Symbols and Mathematical

Convention

Organisation and Presentation in a variety of

representations.

Analysis and Translation of info from one

form-another.

Use of Mathematical Reasoning to develop

coherent sequences. Use of everyday and

mathematical language.

Clear concise justification. of procedures

decisions and/or reasonableness of results.


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Assignment Policy

All assignments must be typed using a word processing program (draft included).Scanned documents will not be accepted (they need to go through TurnIt In to

detect plagiarism).

Drafts must be submitted electronically through TurnIt In, by COB on the due date. Final copies must be submitted electronically via TurnIt In and in hardcopy (either to

the teacher during the lesson or to Tooth if no lesson is on that day), by COB on the

due date.

Failure to submit a final assignment by COB on the due date, will mean that theprogress thus far (i.e. the draft) will be marked and this be grade be awarded.

Student Declaration of Ownership

This is to be complete on submission of the final copy of your assignment.

Students Statement of Authenticity:

I, (name) ________________________, certify that this research and notes are all my ownwork and I have acknowledged all material and sources used in the preparation of thismaterial. Any help received by other people has been acknowledged. I also acknowledgethat I have read the school Assessment Policy and understand the implications of thepolicy.

Signature: Date:

I understand that:

1. Plagiarism is a serious matter and that I may be penalised if this declaration is false.2.Application for extension must be sought before due date except in exceptional

circumstances.

Signature (student): Date:

Signature (parent/guardian): Date:


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Question 1: (2, 2, 1)

The colour of a star is primarily a function of its effective temperature. A star approximates the

behaviour of a black body radiator. As a black body gets hotter, its colour changes (like a stove

hotplate). If you were to heat any solid it would first emit radiation in the infrared region. Further

heating would see it glow a dull reddish colour. With more heating it would glow orange, yellow,

white and eventually blue-hot. Ultimately if it were hot enough a black body emits most of its

energy in the ultraviolet region. Although stars are not perfect black bodies this relationship

between temperature and colour still applies to them.

The table below shows the approximate colour and temperature range (in thousands of degrees

Kelvin, K) of stars in The Milky Way along with an approximation of the numbers of stars of each

colour (they total about 100 billion, 11011

).

Temperature

(1000K)

Colour BlueBlue-

whiteWhite

White-

yellowYellow Orange Red

Number of stars

(billion)6 15 16 19 23 15 6

a) Represent these data in a frequency distribution table. Be sure to include all relevant columns.b) Choose a single, suitable graph type to represent all of these data as effectively as possible. This

means that all data contained in the table above must represented in the same graph. Draw an

accurate graph on grid paper.

c) Calculate the mean temperature and colour of stars.


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Question 2: (1, 1.5, 2.5)

A space agency has determined that, of all trainee astronauts eligible for space flight, 23% will be

over 50 years old before they are selected in a crew. If 12 astronauts are selected at random, what

is the probability that:

A. 3 are over 50?B. At least 10 are over 50? Use your graphics calculator functions to solve this calculation.

You must either describe the functions you are using, or include a screen shot of your

calculator.

C. At most 2 are not over 50?

Question 3: (1, 2, 1.5, 1.5)

The Geometric Distribution is related to the Binomial Distribution in that:

there are two outcomes for each trial success and failure the outcomes for each trial are statistically independent, and all the trials are identical, i.e. they have the same probability of success.

However, where a Binomial random variable is the number of successes in trials, a Geometric

random variable is the number of trials until the first failure. Thus, the total number of trials for a

Geometric distribution is potentially infinite.

A discrete random variable is said to follow a Geometric Distribution if:

( )

( )

Suppose that 20% of items produced by a manufacturing production line are faulty and that a

quality inspector is checking randomly sampled items. What is the probability that:

A. The first item is faulty, i.e. ( )?B. ( )C. ( )D.

( )


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Question 4: MAPS *

Charlotte and Jonathon are playing a game of chance. They are drawing marbles from a bag,

containing 5 pink marbles and 5 blue marbles. Each turn the bag is shaken up and a marble taken

out. If a blue marble is drawn, then Jonathon wins a point. If a pink marble is drawn, then

Charlotte will win the point. The marble is replaced after each turn. They are playing the best of 5

turns.

After 3 turns, Charlotte is ahead 2-1. Jonathon decides that he no longer wishes to play and says

they should split the pot (money that was bet) evenly. Charlotte does not think this is fair.

How should the pot be split so that it is done so with the most fairness? Use calculations to supportyour decision.

State any assumptions you are making, and explain the effect that these will have on your

calculations.


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QUESTIONS 5 7 REFER TO THE DATA SETS GIVEN TO YOU BY YOUR CLASSROOM TEACHER.

Question 5: (2, 3, 3, 3, 3)

In order to improve test results, a nearby school has introduced a new study program for year 11

mathematics students. The Head of Department has asked you to evaluate the effectiveness of the

study program based on one class pre- and post- program test results.

For each of the KPS sections you have been asked to:

A. Construct a frequency distribution table.B. Calculate the mean, median and mode.C. Calculate the range and IQR.D. Calculate the standard deviation.

For both the KPS and CAJ sections, you have been asked to:

E. Draw an appropriate graph.

Question 6: MAPS **

In making your decision about the effectiveness of the study program, you must address the

following items for the three assessable criteria:

i. Measures of central tendency and information on the spread of each of the data sets.ii. Identification of any abnormal or erroneous results and decisions made on their final

inclusion/exclusion.

iii. A decision on the effectiveness of the study program, i.e. based on your statistical analysis,has the study program improved the test results?

Question 7: MAPS *

A new student had joined the class after the pre-program test. Because he had not been there for

both tests, his score was omitted from the analysis. By excluding this student, the standard

deviation of the KAPS section for this data set was reduced by 1.5. Based on this information,

calculate the test score of the new student.


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QUESTIONS 8 and 9 REFER TO THE FOLLOWING SCENARIO.

The behaviour of random variables can be described by a mathematical model of their distribution of

probabilities. A combination of assumptions and data are used to build, check and improve

mathematical models. In fitting a mathematical model, part of the assumptions process is turning

statistical data into probability data. For a discrete variable, the relative frequencies can be used to

estimate probabilities. Being able to express statistical data in terms of a probability distribution can

help make predictions and judgements for future events.

Scenario:

An egg supplier packages eggs into cartons of 6. Preventing damage of any kind during handling,

packing and the retail process is extremely important no one wants to end up with a carton of broken

eggs in their shopping!

Fifty cartons of eggs were selected at random and the number of damaged eggs in each were noted.

The following results were obtained:

# damaged eggs 0 1 2 3 4 5 6

# packets 35 6 4 2 1 0 2

Question 8: (1, 1, JR, 1)

A. Calculate the relative frequencies for the different values of the variable in the above data set.B. Estimate the probability of each of the different values of the variable.C. Is it possible to fit the data to a probability distribution? Use your knowledge of discrete

random variables and probability distributions to explain your answer.

D. Graph the probability distribution. The graph must be hand drawn and suitable graph papermust be used.


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Question 9: MAPS ***

A. It may be possible to fit the data to a Binomial Distribution. What specific assumptions need tobe made in this scenario in order for the Binomial Distribution to be the right model for the

situation?

B. Using your estimated probabilities, find a possible Binomial Distribution model.C. Compare the theoretical model with the data. Is the Binomial Distribution a good fit for this

data? Explain your answer.


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APPENDIX 1

Outliers:

An outlier is a result that lies outside of the overall pattern of a distribution. For discrete data, an

outlier can be defined as a result that falls more than 1.5 times the IQR above the third quartile, or

below the first quartile.

11mab_-_2010_-_assessment_2.2_-_empt[1]

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