Exam 1 2016, questions and answers PDF

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Study!Period!5!Exams,!2016!

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MATH1053!Quantitative!Methods!for!Business!

Student!Number!

|__|__|__|__|__|__|__|__|__|!

Last!Name!

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_______________________ !

First!Name!

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_______________________ !

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! This!exam!paper!must!not!be!removed!from!the!venue!

Make! _________________! Model

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Information!Technology!and!Mathematical!Sciences! EXAMINATION!SOLUTIONS! MATH1053!Quantitative!Methods!for!Business! This%paper%is%for%City%West,%External%Central%Exam%Venue%(External)%and%External%Exam%Venue% (External)%students.% %% Examination!Duration:! !

120!minutes!

Reading!Time:! !

10!minutes!

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Exam!Conditions:! #

Extra!time!and!bilingual!dictionary!provision!allowed!for!ENTEXT!eligible! students!

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This!exam!is!part!open!book! !

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Materials!Permitted!In!The!Exam!Venue:!

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(No!electronic!aids!are!permitted!e.g.!laptops,!phones)! Any!calculator!permitted!

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Materials!To!Be!Supplied!To!Students:!

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1!x!Number!of!Attendance!Slips!for!School!per!student!

For#Examiner#Use#Only:#

Instructions!To!Students:!

Question#

Mark#

Max. Mark

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15)

2)

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14)

3)

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22)

4)

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17)

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16)

6)

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16)

Total#

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100#

•! It#is#not#permitted#to#bring#books,#notebooks#or#notes#into#the#exam.# •! Graphics) calculators) must) be) cleared) by) the) candidate) upon) request,) during)ID)and)other)checks)in)the)exam.) •! There)are)six)(6))questions.)Marks)are)given)for)each)question.# •! A)total)of)100)marks)constitutes)full)marks)for)this)examination.# •! Candidates)should)not)use)red)or)green)pens)to)write)in)the)exam)paper.# Attachments#include#future#and#present#value#tables,#the#standard#Normal# table,#the#Mathematical#formulae#and#the#Statistical#formulae.#These#can# be#removed#for#the#duration#of#the#exam#but#must#be#handed#in#to# invigilator#together#with#the#completed#exam#paper.

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Question 1 (Total of 15 marks)

MATH1053!Quantitative!Methods!for!Business!

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Ellen B runs the popular YouTube fitness program Do Sweat It! for thousands of online exercise students from around the world. She now wants to open a studio in her local city to offer in-person exercise classes. She has located a studio she wishes to rent however she needs $80,000 to refurbish it before she can open for business. (a)!(6 marks)! Ellen B tried crowdfunding and raised 35% of her $80,000 target. She wants to deposit this crowdfunded amount into a bank account for 6 months with an interest rate of 4.5% per annum, compounded quarterly. How much will Ellen B have at the end of 6 months? Will she have at least $30,000 to put towards her target amount? Solution:!

! = 0.35×80000 = 28000! * +=

0.045 = 0.01125! 4

* .=

6 ×4 = 2 12 (1#+#1#+#1#=#3#marks)# 0 = ! 1 + + 2* = 28000× 1 + 0.01125 3 * = 28,633.54 (1#+#1#=#2#marks)#

Ellen B will have $28,633.54 in 6 months’ time. She will not have at least $30,000 to put towards her target. (0.5#+#0.5#=#1#mark)# (b)!(7 marks)! Ellen B has realised she needs to take out a loan to refurbish the studio. Combining her savings with the crowdfunded money, she has a deposit of $38,000 to put towards her goal of $80,000 and needs a loan for the remaining balance from a lender. Ellen B has opted for a 2-year repayment plan, with an interest rate of 7.5% per annum compounded monthly. What size monthly repayments will Ellen B need to make? Solution:!

5 = 80000 − 38000 = 42000* 0.075 = 0.00625* += 12 . = 2×12 = 24

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MATH1053!Quantitative!Methods!for!Business!

(1#+#1#+#1#=#3#marks)#

8=

5 1− 1++ +

92

=

42000 1 − 1 + .00625 . 00625

93:

=

42000 = $1,889.98* 22.22242

(1#+#1#+#1#=#3#marks)# Ellen will need to make monthly repayments of $1,889.98)

)

)))))))))))(1#mark)#

(c)!(2 marks)!Ellen B is contacted by a new lender offering the same loan interest rate of 7.5% per year with fortnightly compounding. Compared to the lender in (b) explain which option is to Ellen’s financial advantage. Do not perform any calculations. Solution:!

Since the compounding frequency is more often than her current loan, the amount of interest Ellen B will pay will be more. Ellen B should stay with her current loan. (2#marks)# Any#sensible#answer#quoting#impact#of#compounding#frequency## +#Ellen#staying#is#OK

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MATH1053!Quantitative!Methods!for!Business!

Question 2 (Total of 14 marks) The Luck of the Irish is a phrase often used to refer to unexpected good luck. To investigate whether there is any truth to this phrase, a study examining the luck of Irish and Non-Irish people was conducted in a casino, with participants playing a game called Pot of Gold. The winnings of the players were classified as Empty Pot (participants won nothing), Dram (participants won amounts up to $500) and Pot of Gold (participants won amounts of at least $500). The results were summarised as below: Winnings Nationality

Empty Pot

Dram

Pot of Gold

Total

Irish

28

116

11

155

Non-Irish

62

101

2

165

Total

90

217

13

320

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)

(a)!(2 marks) What is the probability a randomly selected participant is Irish and won a Pot of Gold? [Include an appropriate probability statement.] Solution:!

P)(Irish)and)Pot)of)Gold))=)

== >3?

))=)0.0344)or)3.44%)

) (1#+#1#=#2#marks)#

(b)!(2 marks) What is the probability a randomly selected participant is Non-Irish or the winnings were a Dram? [Include an appropriate probability statement.] Solution:!

P)(Non)Irish)or)Dram))=)

@3A=?=A3A==@ 3B= >3?

)=)

>3?

))=)0.8781)or)87.81%) ) (0.5#+#1#+#0.5=#2#marks)#

(c)!(3 marks) Suppose that a player’s winnings are an Empty Pot. What then is the probability this person is Irish? [Include an appropriate probability statement.] Solution:!

P)(Irish)|)Empty)Pot))=)

3B C?

))=)0.3111)or)31.11%) )

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))(1#+#1#+#1#=#3#marks))

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MATH1053!Quantitative!Methods!for!Business!

(d)!(7 marks) Are Irish and Winnings statistically independent? Show all calculations that support your answer. Based on the calculated probabilities, is there any truth to the phrase “Luck of the Irish”? [Include appropriate probability statements.] Solution:!

P(Irish) =

=DD >3?

))=)0.4844)or)48.44%)

P)(Irish)|)Empty)Pot))=)0.3111)from)(c)) P)(Irish)|)Dram))=)

==@ 3=E

))=)0.5346)or)53.46%)

P)(Irish)|)Pot)of)Gold))=)

)

== =>

))=)0.84.62)or)84.62%)) (1#+#1#+#1#+1#=#4#marks))

Since)the)probabilities)differ)Irish)and)Winnings)are)statistically)dependent.))) ))))))(1#+#1##=#2#marks)# Based) on) this) analysis) I) think) there) is) some) truth) to) the) phrase) (especially) as) they) won)nearly)85%)of)the)Pot)of)Gold)category!).) ) )) ))))))) 1#mark## (anything#sensible#based#on#the#analysis)#

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MATH1053!Quantitative!Methods!for!Business!

Question 3 (Total of 22 marks) Olivia is a University student with an online jewellery store called All That Glitters, where she sells earrings and necklaces. She is hoping to use the money earned from her online store to fund an overseas holiday at the end of the year. As Olivia is a student she has only 10 hours each week to make jewellery. She knows that it takes her 30 minutes to make a pair of earrings and 1 hour to make a necklace. It takes Olivia 5 grams of metal to make earrings and 20 grams to make a necklace, however Olivia has at most 250 grams of metal to use each week. She has a total budget of $80 each week to spend on materials; each pair of earrings costs $2 to make and each necklace costs $6 to make. As the current fashion trend is to wear multiple necklaces, Olivia would like to makes at least 8 necklaces each week to meet this fashion demand. For each pair of earrings sold Olivia will make $15 profit while each necklace will make a $20 profit. How many necklaces and pairs of earrings should Olivia make each week to maximise profit, assuming she sells all jewellery? For the questions that follow, use the following notation for the decision variables: E)=)Number)of)Earrings)sold)weekly) N)=)Number)of)Necklaces)sold)weekly) The Excel Solver output given on the next page may be of assistance when answering questions (a) – (e). (a)!(2 marks) Write an equation for the time constraint in this model. ! Solution:!

0.5E + N ≤ 10

1#mark#for#LHS# 0.5#for#≤# 0.5#marks#for#RHS## #

(b)!(2 marks) Name the two model constraints that form the optimal solution to this model. Briefly explain how you know. Solution:!

The constraints for Time and Fashion Demand form the optimal solution to this model 1#mark# as these two constraints are binding. 1#mark#(mark#leniently)#

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MATH1053!Quantitative!Methods!for!Business!

Microsoft!Excel!14.1!Answer!Report! Objective!Cell!(Max)! Cell! $B$13!

Original! Value!

Name!

Profit!!

!$275.00!!

Final!Value!

!$220.00!!

Variable!Cells! Cell! $B$4! $C$4!

Original! Value!

Name! E! N!

Final!Value!

5! 10!

Constraints! Cell! Name! $B$17! Time!LHS! $B$18! Metal!LHS! $B$19! Budget!LHS! ! $B$20! Fashion!Demand!LHS! $B$21! Nonaneg.!E!LHS! $B$22! Nonaneg.!N!LHS!

Cell!Value! 10! 180! 560! 8! 4! 8!

4! 8!

Formula! $B$17=$D$22!

Status! Binding! Not!Binding! Not!Binding! Binding! Not!Binding! Not!Binding!

Slack! 0! 70! 24! 0! 4! 8!

Peter: this was a typo – please see my instructions for (c) Microsoft!Excel!14.1!Sensitivity!Report! Variable!Cells! !! Cell! $B$4! E! $C$4!

!! Name!

N!

Constraints! !! !! Cell! Name! $B$17! Time!LHS! $B$18! Metal!LHS! $B$19! Budget!LHS! $B$20! Fashion!Demand! $B$21! Nonaneg.!E!LHS! ! $B$22! Nonaneg.!N!LHS!

Final! Reduced! Objective! Allowable! Value! Cost! Coefficient! Increase! 4! 0! 15! 1E+30! 8!

0!

20!

10!

Final! Value!

Shadow! Constraint! Allowable! Price! R.H.!Side! Increase! 10! 30! 10! 6! 180! 0! 250! 1E+30! 56! 0! 80! 1E+30! 8! a10! 8! 2! 8! 0! 0! 8!

8!

0!

Page!7!of!19!

0!

8!

Allowable! Decrease! 5! 1E+30!

Allowable! Decrease! 2! 70! 24! 8! 1E+30! 1E+30!

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MATH1053!Quantitative!Methods!for!Business!

(c)! (4 marks) Describe the linear programming solution in terms of: •! The number of pairs of Earrings and Necklaces that should be sold each week to maximise profit. •! The maximum profit per week. •! The resources that will be fully used each week to make the jewellery •! The resources that will not be fully used each week to make the jewellery) ) Peter:!note!there!was!an!errant!0!in!the!answer!report!(560!instead!of!56!for!the! budget).!!Please!mark!correct!if!students!refer!to!either!the!$80!budget!or!the!larger! budget!using!$560.!!I’ve!highlighted!the!issue!in!the!answer!repot.! ! Solution:!

From the Answer Report:

(0.5#marks)

Olivia should sell 4 pairs of Earrings and 8 Necklaces each week to maximise profit. (1#mark) The maximum profit per week will be $220.

))))))))))))))))))))(0.5#marks)#)

All of Olivia’s time will be used each week and the Fashion Demand will be met (binding constraints) (1#mark)## Olivia’s budget and metal materials will not be fully utilised (not binding)

)(1#mark)##

) (d)!(4 marks) Olivia would like to increase prices to yield a higher profit per necklace. If Olivia’s profit made by selling necklaces increases to $28, will the optimal solution in part (c) still be optimal? Give reasons for your answer and calculate the new maximum weekly profit if appropriate.) ) Solution:!

From the Sensitivity Report:

(0.5 marks)

The allowable increase for profit from Necklace sales is $10.

(0.5 marks)

Since the increase from $20 to $28 is within the allowable increase range, the (1 mark) quantities will stay the same. The max profit per week will become Profit = 15E + 28N = 15x4 + 28x8 = $284 OR Profit per week will increase by 8 × $8, i.e. 220 + 64 = $284.

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MATH1053!Quantitative!Methods!for!Business!

An answer that correctly discusses allowable range from the Sensitivity Report, indicates no change in quantities, and gives the expected change in income is OK.

(e)! (4 marks) Olivia has reduced her other commitments so she can spend more time making jewellery to sell. She would now like to spend 15 hours each week making jewellery. Will the optimal solution from part (c) change and will Solver need to be re-run? Find the new value of maximum weekly profit if appropriate (hint: you will need to use the shadow price). Solution:!

From the Sensitivity Report,

(0.5 marks)

The shadow price for Time is $30 and the range of feasibility for this constraint is (10-2, 10+6) (8,16) (1 marks) Since 15 hours is within the range of feasibility, we can use the shadow price to find (1 mark) the new maximum weekly profit. The maximum profit will increase by 30 × 5 = $150 per week from $220 to $370 per week. (1 mark) The optimum quantities of Necklaces and Earrings will change as well (run Solver (0.5 marks) again). Any# answer# that# explicitly# and# correctly# discusses# the# shadow# price# from# the# Sensitivity#Report,#and#gives#the#expected#change#in#profit#is#OK.#

(f)! (6 marks) Olivia is considering changing her business model to sell only necklaces and would like to sell each necklace for $35. She has estimated her weekly fixed cost to be $120 with weekly variable costs of $18. Calculate the number of necklaces Olivia should sell each week to break even. Based on this break-even quantity, will Olivia meet the fashion demand of at least 8 necklaces per week? Finally, if Olivia’s variable costs were to increase due to increased materials costs, will the break-even quantity increase, decrease, or stay the same? Explain briefly but do not perform any calculations (hint: consider the role of the contribution margin). Solution:!

s)=35) ))f)=)120) )v)=)18) )

)

)

)

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))))(1.5#marks))

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MATH1053!Quantitative!Methods!for!Business!

)x)=)?) ) G= )) )

)

)

)

H 120 = = 7.06) I − J 35 − 18 )

)

)

)

))))(0.5#+#0.5#=1#mark)#

Olivia needs to sell 8 necklaces per week to break even. Yes she will (just) meet the minimum fashion demand!##

)))))))(1#mark)# #

######(0.5#marks)#

If the variable costs increase, the contribution margin s-v per necklace sale will decrease, meaning that Olivia will need to sell more necklaces to break-even. # # # # # # # 2#marks#(mark#leniently#–#any#sensible#answer#OK)

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MATH1053!Quantitative!Methods!for!Business!

Question 4 (Total of 17 marks) Are we spending too much too soon? A recent study investigated the amount of debt accumulated in the first year of having a credit card. A sample of 72 Visa-card holders, who opened their accounts one year ago was taken, with the data summarised as below.

1000

2000

3000

4000

5000

Credit'Card'Debt'($)'' Mean 4326.39 Standard Error 102.34 Median 4462.50 Mode 4113.75 Q1 3996.56 Q3 4900.31 Std Deviation 868.42 Sample Variance 754,151.69 Kurtosis 3.21 -1.48 Range 4597.50 Minimum 922.50 Maximum 5520 Sum 311,499.75 Count 72

Histogram7of7Credit7Card7Debt 40 Frequency

Credit'Card'Debt'($)

30 20 10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Credit7Card7Debt7($)

Nick: the histogram is a trouble-marker here ! Please mark leniently – most students should have used the Maximum value in the table but if they referred to the max in the histogram, please mark as correct ! (a)! (5 marks) Describe the and of the distribution. Verify the existence of outliers using the 1.5xIQR rule and using the boxplot, provide the exact number of outliers in this distribution. [Hint: you will need to compute the Interquartile Range from the output above for the outlier test.] Solution:!

Approximately symmetric

(0.5 marks)

One major peak

(0.5 marks)

IQR = Q3 - Q1 = 4900.31 – 3996.56 = 903.75

(0.5 marks)

Using the 1.5xIQR rule we have: Q1 – 1.5xIQR = 3996.56 – 1.5x903.75 = 2,640.94

(1 mark)

Q3 + 1.5xIQR = 4900.31 + 1. 5x903.75 = 6,255.94

(1 mark)

Since the min value is s

there is (0.5 marks - mark leniently)

From the

the

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(1 mark)

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MATH1053!Quantitative!Methods!for!Business!

(b)! (4 marks) Which measures of central tendency and dispersion should be used to describe this distribution? Give (brief) reasons. Give and interpret their values. Solution:!

Since the distribution has outliers, the median should be used as a measure of central (1.5 marks) tendency and IQR as a measure of dispersion (for consistency). Median = 4462.5

(0.5 mark)

A typical credit card debt is $4,462.50. Half of the credit card debts are less than (1 mark) $4,462.50 and half are more than $4,462.50. From (a), IQR = 903.75 thus credit card debt was $903.75, (1 mark) (c)! (6 marks) A ‘get-out-of-debt’ report aimed at reducing credit card debt claimed that the average credit card debt of first-time card holders is $4000 with a standard deviation of $850. Test at the 0.05 level of significance whether there is evidence that the average credit card debt in part (a) is different from this reported amount. Solution:!

Step 1

H0: µ)=4000) H1: µ...