QBUS5001:QUANTITATIVEMETHODSFORBUSINESS ASSIGNMENT

Question1(10marks)

A business analyst is interested in studying the weekly range data in stock prices of twoAustralian banks: ANZ bank and NAB bank. The weekly range is defined as the highest stockprice minus the lowest stock price in a week. He collects weekly range data from 2015 to 2019and thedata aregivenin SpreadsheetQ1. Summarystatisticsaregiven below.

• Instruction:Go to Spreadsheet Q1. Perform statistical analysis on the data and answerthefollowingGiveyouranswersto2decimalplacesunlessspecified.

• [1mark]Howmanyweeklydatathatthebusinessanalysthascollected?Giveyouranswerasan integer.
• [2 marks] Is it possible that ANZ range data and NAB range data have a linearrelationship?Youmaydrawascatterplottohelpyouanswerthis(Yes/No)
• [2 marks] What is the correlation coefficient between the range data of these twostocks?
• [1mark]WhatisthemeanweeklyrangeofANZstockprice?
• [1mark]WhatisthestandarddeviationoftheweeklyrangeofNABstockprice?
• [1 mark] Is the distribution of the weekly range of the two stocks skewed to the left?(Yes/No)
• [2marks]DoesNABhaveasmallercoefficientofvariationcomparedtoANZ?(Yes/No)

Question2(10marks)

A department store analysed its most recent sales and determined the relationship betweenthe way the customers paid for the item and the way they shop. The joint probabilities and themarginalprobabilitiesare given inthe table below.

Answerthefollowingquestionsandgiveyouranswersto2decimalplaces.

• [1mark]Whatproportionofpurchaseswaspaidbycreditcardandwascompletedonline?
• [1mark]Whatproportionofpurchaseswaspaidbyanon-cashmethodandwascompleted in store?

• [1mark]Whatproportionofpurchaseswaspaidbydebitcard?
• [1mark]Whatproportionofpurchaseswascompletedonline?
• [2marks]Findtheprobabilitythatapurchasewascompletedonlinegiventhatthispurchasewaspaidbydebit card.
• [2marks]Findtheprobabilitythata purchasewaspaidbycashgiventhatthispurchasewascompleted
• [2marks]AretheeventsModeofShoppingandMethodofPaymentindependent?(Yes/No)

Question3(20marks)

Let X and p be the number of heads and the proportion of heads, respectively, in a binomialexperience of tossing eighty identical and unfair coins. Obviously, X ~ Bin(n, p) where n = 80andpisunknown.Thisbinomialexperimentisrepeatedindependentlyfor1000timesandthedataaregiveninSpreadsheetQ3.Foreachexperiment,thesampleproportion ???? iscalculated.

• Instructions: Go to Spreadsheet Q3. Perform statistical analysis on the data andanswerthefollowingGiveyouranswersto4decimalplacesunlessspecified.

• [2marks]Findthe90thpercentileoftheproportionof
• [2marks]Find thethirdquartileoftheproportionof
• [2marks]FindtheIQRoftheproportionof
• [2marks]Find CVoftheproportionof
• [4 marks] How many outliers are there in these 1000 values of???? ? Give your answer asan
• [1mark]Isthedistributionof????morelikelytobesymmetric?(Yes/No)
• [1 mark] Can the Central Limit Theorem be applied to approximate the samplingdistributionfor?????(Yes/No)

Answerthefollowingquestionsbasedontheoutcomefromthefirstbinomialexperiment.

• [2marks]Whatisthe standard errorof?????

• [2marks]Whatisthelowerconfidencelimitofa99%confidenceintervalforp?
• [2marks]Whatistheupperconfidencelimitofa99%confidenceintervalforp?

Question4(20marks)

FindthefollowingprobabilitiesusingExcel.Giveyouranswerto4decimalplaces.

• [3marks]Find P(15<X25)whereX~Poi(20).
• [3marks]Find P(3X<6)whereX~Exp(0.25).
• [4 marks] A random sample of size 25 is taken from the normal N(70, 100) distribution.Find theprobabilitythatthe averageofthese25valuesisgreaterthan
• [5marks]Forabinomialexperimentwithn=150andp=60,usethenormalapproximation to find the probability that the number of successes, X, is between 84 and99, inclusive.
• [4 marks] For a binomial experiment with n = 150 and p = 0.60, find the probability thatthesampleproportionofsuccess????isbetween56and 0.66, inclusive.
• [1mark] Istheanswerin(v)exact?(Yes/No)

Question5(15marks)

A training course claims that it can increase an adults intelligence quotient (IQ) score by mpoints. A business analyst is interested in studying this claim. She takes a random sample of200 adults who participated in this training course and obtains 200 increases in the IQ score.Data are saved in Spreadsheet Q5. She wants to construct a confidence interval for m. It isassumedthattheincreaseinIQscorefollowsa normaldistributionwithunknownvariance.

• Instruction: Go to Spreadsheet Q5. Perform statistical analysis on the data andanswer the following questions. Give your answers to 2 decimal places unless

• [2marks]Whatisthepointestimateofm?
• [2 marks] What is the z-value or t-value required for the calculation of a 95% confidenceintervalform?

• [2marks]Whatisthelowerconfidencelimitofa95%confidenceintervalform?
• [2marks]Whatistheupperconfidencelimitofa95%confidenceintervalfor m?
• [1mark] Istheupperconfidencelimitofa95%confidenceintervalformsmallerthanthatofa 90% confidence interval?(Yes/No)
• [1mark] Isthelowerconfidencelimitofa99%confidenceintervalformsmallerthanthatofa 95% confidence interval?(Yes/No)
• [5 marks] If the business analyst requires that the difference between the populationvalue and the sample value of the increase in IQ score to be at most 0.5 point with a 95% What is the minimum sample size that she should take? From a very similartrainingcourse, therangeofincreaseis14points. Youranswermustbeaninteger.

Question6(10marks)

A business analyst wants to study the impact of Coronavirus outbreak on peoples onlineshopping habits in Sydney. She collects a random sample of 81 households from an onlinesurvey and obtains the weekly expenditures on online shopping before and after theoutbreak. After analysing the data, she finds that the average increase in weekly expenditureon online shopping is $250 with a standard deviation of $75. The business analyst isinterested in estimating the mean, $m, of the increase in weekly expenditure on onlineshopping. Fornumericalquestions, give youranswers to2decimalplaces.

• [1mark] Isthisamatchedpairexperiment?(Yes/No)
• [1mark] CantheCentralLimitTheorembeappliedtothissituation?(Yes/No)
• [1 mark] Shall we assume that the population standard deviation of the increase inweeklyexpenditure isunknown?(Yes/No)
• [5marks]Findthewidthofa99%confidenceintervalform.
• [1mark] Isa99%confidenceintervalformwiderthana90%confidenceinterval?(Yes/No)
• [1 mark1] Suppose that the population standard deviation is known to be $75. Will a99%confidenceintervalfor mbewiderthana99%confidenceintervalin(iv)?(Yes/No)

Question7(15marks)

Many Australians who buy a property need a mortgage from a leader which is a financialinstitution. A borrower can obtain a home loan directly from a lender or indirectly via amortgage broker. A business analyst wants to study whether borrowers are satisfied with theservice provided by the lenders and mortgage brokers during their home loan applicationprocess. Let p1 be the proportion of borrowers who obtain a home loan directly from a lenderandaresatisfiedwiththeserviceprovidedbythelenders,andp2betheproportionofborrowers who obtain a home loan via a mortgage broker and are satisfied with the serviceprovided bythe brokers.

The business analyst takes a random sample of 80 borrowers who directly obtain their homeloan from lenders and 64 of them are satisfied with the service provided. In another randomsample of 60 borrowers who obtain their home loan via mortgage brokers, 45 of them aresatisfiedwiththeserviceprovided.Forthefollowingquestions,giveyouranswersto4decimalplaces.

• [1mark] Arethetwosampleslikelytobeindependent?(Yes/No)
• [1mark]Whatisthebestunbiased estimateofp1p2?
• [3marks]Whatisthestandard erroroftheestimateofp1p2?
• [3marks]Whatisthelowerconfidencelimitofa99%confidenceintervalforp1p2?
• [3marks]Whatistheupperconfidencelimitofa95%confidenceintervalforp1p2?
• [4 marks] Now, suppose that the business analyst wants to estimate p1 p2within amargin of error of 0.05 from the true difference with 90% confidence. Suppose that hehasnopriorknowledgeaboutthevaluesofp1andp2.Findtheminimumsamplesize,n1

= n2= n (say) that he needs to take from each population. Your answer must be aninteger.