QBM117 Business Statistics Assignment 2 Due date: Sunday, 01 January, 2017 Value: 12% Rationale Assignment 2 is designed to assess the following learning outcomes: that students will  be able to calculate and interpret probabilities, and use standard discrete and continuous probability distributions;  be able to use a statistical package to analyse data appropriately, and then interpret the output;  be able to evaluate whether the assumptions underlying statistical techniques are valid in a given scenario.  be able to apply basic principles of survey design, such as determination of appropriate sample sizes and sampling techniques. Presentation The assignment must be neatly handwritten with any Excel output inserted where required at the appropriate place in the assignment not in an Appendix at the back of the assignment. Mathematical formulae and notation is too difficult to type up. Marks will be deducted for assignments which do not follow these guidelines. The assignment is to be uploaded to EASTS as a single Word or PDF file. PDF files maintain the integrity of the document when images are included so you may prefer to upload a PDF file rather than a Word file. Assignments submitted in non-printable formats such as a ZIP file or as a collection of images will not be marked. If your scanner produces separate graphics files please paste them into a Word document before submitting to EASTS. Pages must be numbered, and your name and student number must be included on every page. Please view your uploaded assignment in EASTS to ensure all pages, graphs, images have uploaded correctly. If they have not, then please review the format of your assignment and resubmit. A PDF file retains the integrity of the document so you might consider submitting as a PDF file. Marking Criteria: The marking criteria for each question are given in a table immediately below each question. Please read these tables carefully so you know what the marker is expecting.The grade that you receive for this assignment as a whole is determined by the cumulative marks gained according to the tables below (FL 0-49%; PS 50-64%; CR 65-74%; DI 75-84%; HD 85-100%).Question 1 (20 marks) a. If P A B P A P B ( ) 0.08, ( ) 0.2 and ( ) 0.4,     then i. calculate P A B ( | ) ii. state whether the events A and B are independent or not iii. justify the decision you made in ii. (4 marks) Marking Criteria 0 marks 1 mark 2 marks a.i. Formula/working incorrect or not provided; incorrect answer Correct formula and working or correct answer (no working) Correct formula and working and correct answer a.ii. Incorrect answer Correct answer a.iii. Incorrect or irrelevant justification Relevant justification b. A coffee shop manager randomly selects 40 of the patrons of his coffee shop. He surveys them about their coffee and tea drinking habits. Twenty of them drink coffee, fifteen of them drink tea and five drink both. i. How many of them drink neither tea nor coffee? If C is defined as the event that a patron drinks coffee and T as the event that a patron drinks tea, ii. calculate P C ( ) . iii. calculate P T C ( | ) and write a sentence that describes this probability in terms of the question. iv. calculate P T ( ) and write a sentence that describes this probability in terms of the question. (8 marks) Marking Criteria 0 marks 1 mark 2 marks 3 marks b.i. Incorrect answer Correct answer b.ii. Incorrect answer Correct answer b.iii. None of three aspects listed under 3 marks correctly provided. One out of three aspects listed under 3 marks correctly provided Two out of three aspects listed under 3 marks correctly provided Correct answer; relevant working; clearly expressed description of probability b.iv. None of three aspects listed under 3 marks correctly provided. One out of three aspects listed under 3 marks correctly provided Two out of three aspects listed under 3 marks correctly provided Correct answer; relevant working; clearly expressed description of probabilityc. One of the questions in a recent household survey asks respondents to count the total number of mobile phones owned by members of the household. The table which follows shows the probability distribution for the number of mobile phones per household. X 0 1 2 3 4 5 6 7 8 or more p(x) 0.00 0.02 0.18 0.30 0.20 0.15 0.10 0.05 0.00 i. Discuss how we can confirm that this is a valid probability distribution. ii. Calculate the mean number of mobile phones owned per household. iii. Calculate the standard deviation number of mobile phones owned per household. iv. What is the probability that for a randomly selected household, the number of phones in this household is at least 3? (8 marks) Marking Criteria 0 marks 1 mark 2 marks 3 marks c.i. No relevant justification One relevant statement Two relevant statements c.ii. Neither of two aspects listed under two marks correctly provided One out of two aspects listed under two marks correctly provided Answer correct with relevant working including formula c.iii. None of three aspects listed under 3 marks correctly provided. One out of three aspects listed under 3 marks correctly provided Two out of three aspects listed under 3 marks correctly provided Correct Formula; relevant working; Correct answer c.iv. Incorrect Answer. Correct AnswerQuestion 2 (12 marks) 11+1 for sentences which answer the questions in parts c. and d. A manufacturing company uses two bottling machines to produce 2 litre bottles of apple juice. Machine A can bottle 200 bottles of juice per hour. Machine B can bottle 300 bottles per hour. Their quality control process has found that if a bottle of juice has been bottled by Machine A, then the probability that it has been under-filled is 4.5%. If a bottle of juice has been bottled by Machine B, however, then the probability that it has been under-filled is 6%. a. Using letters of the alphabet, define the two simple events described in this problem and also their complements. (4 events altogether). Probability notation is not required to do this. (2 marks) b. Draw a probability tree to represent the information given in the question using the same letters that you used to define the simple events in part a. Note: Marks will be deducted if the probabilities do not appear along the branches of the probability tree. (3 marks) c. Determine the probability that a randomly chosen bottle contained the right amount of juice. (3 marks) d. Given that a randomly selected bottle was not under-filled, what is the probability that the bottling was done by Machine B? (3 marks)Marking Criteria 0 marks 1 mark 2 marks 3 marks a. Fewer than 2 definitions 2 or 3 definitions using letters of the alphabet 4 definitions using letters of the alphabet b. No probability tree Probability tree with several flaws Correctly labelled probability tree with some probabilities missing from branches of the tree or incorrect Correctly labelled probability tree with correct probabilities included along all branches of the tree c. 0 of the 3 aspects listed under 3 marks 1 of the 3 aspects listed under 3 marks 2 of the 3 aspects listed under 3 marks Correct translation of question into a mathematical probability statement and formula; correct numerical calculations and answer d. 0 of the 3 aspects listed under 3 marks 1 of the 3 aspects listed under 3 marks 2 of the 3 aspects listed under 3 marks Correct translation of question into a mathematical probability statement and formula; correct numerical calculations and answer Sentences Sentence answering the question included for one or fewer parts Sentence answering the question included for parts c. and d.Question 3 (9 marks) 8+1 for sentences which answer the questions in parts b. and c. A manufacturer vacuum cleaner parts knows that for every 1000 units produced, 4 units are faulty. The manufacturer wants to estimate, using probabilities, the likelihood of faulty units in a selection of 500 randomly selected units. a. Write down the names (using single letters or Greek symbols) of the two parameters of this binomial distribution and their values. (2 marks) b. Use the appropriate probability formula to calculate the probability that there are no faulty components among these units. (3 marks) c. Use Excel to calculate the probability that there are more than 10 faulty units among these units. Express your answer as a decimal correctly rounded to 6 decimal places. (3 marks) Marking Criteria 0 marks 1 mark 2 marks 3 marks a. Parameters named incorrectly and values incorrect Both parameters named correctly or both values correct or one parameter and one value correct Parameters named correctly; parameter values correct B 0 of the 3 aspects listed under 3 marks provided correctly 1 of the 3 aspects listed under 3 marks correct 2 of 3 aspects listed under 3 marks correct Correct translation into probability statement along with correct substitution into formula and correct answer c. 0 of the 3 aspects listed under 3 marks provided correctly. 1 of the 3 aspects listed under 3 marks correctly provided. 2 of 3 aspects listed under 3 marks correctly provided. Correct translation of 'more than' into a mathematical inequality along with correct Excel formula and correct answer correct to 6 decimal places Sentences Sentence answering the question included for one or fewer parts Sentence answering the question included for parts b. and c.Question 4 (9 marks) 8+1 for sentences answering the questions A shire council has found that historically 20 % of ratepayers choose to pay their rates by instalments. A sample of 25 new ratepayers to the shire were asked if they had paid their current rates by instalment. a. Write down the name of the distribution being described in this problem and write down the name/s of its parameter/s (using single letters or Greek symbols) and its/their values. (2 marks) b. Using the tables appropriate for this distribution, calculate the probability that fewer than a quarter of those new ratepayers sampled had paid their rates by instalment. (3 marks) c. Using Excel, calculate the probability that more than 5 but fewer than 12 had paid their rates by instalment. (3 marks) Marking Criteria 0 marks 1 mark 2 marks 3 marks a. Incorrect distribution and at least one parameter(s) missing Distribution incorrect or at least one parameter(s) missing Correct distribution and parameter(s) b. 0 of the 3 aspects listed under 3 marks 1 of the 3 aspects listed under 3 marks 2 of 3 aspects listed under 3 marks Correct translation of 'fewer than' into a mathematical inequality; correct value of X in the inequality and correct answer. c. 0 of the 3 aspects listed under 3 marks 1 of the 3 aspects listed under 3 marks 2 of 3 aspects listed under 3 marks Correct translation of 'more than…but fewer than…' into a mathematical inequality along with correct Excel formula and correct answer Sentences Sentence answering the question included one or fewer parts Sentence answering the question included for parts b. and c.Question 5 (9 marks) 8+1 for sentences answering the questions The manager of a group of suburban service stations has noticed a high level of driveoffs at the busiest of these service stations. Using CCTV footage, the manager estimates that the number of drive-offs on a Saturday night is 4 on average. (A driveoff involves a driver driving off without paying for petrol). a. Write down the name of the probability distribution being described and write down its parameter/s (using letters or Greek symbols) and its/their value/s. (2 marks) b. What is the probability that there is no more than 1 drive-off on a given Saturday? (2 marks) c. Use Excel to evaluate the probability that there are more than a total of 20 drive-offs on all Saturdays in the month of December 2016. Take care here. (4 marks) Marking Criteria 0 marks 1 mark 2 marks 3 marks 4 marks a. Incorrect distribution and at least one parameter(s) missing Distribution incorrect or at least one parameter(s) missing Correct distribution and parameter(s) b. 0 of the 2 aspects listed under 2 marks 1 of the 2 aspects listed under 2 marks Correct translation of 'no more than' into a mathematical inequality along with correct answer using Statistical Tables c. 0 of the 4 aspects listed under 4 marks 1 of the 4 aspects listed under 4 marks 2 of 4 aspects listed under 4 marks 3 of 4 aspects listed under 4 marks Correct translation of 'more than' into a mathematical inequality; relevant working with correct use of Excel formula; correct parameter substitution into formula; correct answer. Sentences Sentence answering the question included for one or fewer parts Sentence answering the question included for parts b. and c.Question 6 (16 marks) 15+1 for sentences answering the questions b., c. and d. A company pays the staff the award rate. Some of its employees work over-time. For employees employed under a particular award, the employees receive on average $1025 per week with a standard deviation of $250 per week. a. Write down the names of the parameters of this normal distribution(using Greek symbols) and their values. (2 marks) b. A randomly chosen employee working under this award is selected. Calculate the probability that that he earned less than $1400 the previous week. (4 marks) c. Twenty staff employed under this same award are randomly selected. What is the probability that the average weekly pay the previous week for this sample of twenty staff exceeded $1150? (5 marks) d. A randomly chosen employee working under this award is selected. There is an 80% chance that this employee earns more than $a. What weekly income is represented by the letter 'a'? (4 marks)Marking Criteria 0 marks 1 mark 2 marks 3 marks 4 marks 5 marks a. Neither parameter value is correct Only one parameter value correct Both parameter values correct b. 0 of the 4 aspects listed under 4 marks correct 1 of the 4 aspects listed under 4 marks correct 2 of the 4 aspects listed under 4 marks correct 3 of the 4 aspects listed under 4 marks correct Correct translation of 'less than' into a mathematical inequality along with relevant fully labelled diagram; correct transformation of probability and numerical calculations. c. 0 marks if incorrect distribution chosen. 1 of the 5 aspects listed under 5 marks 2 of 5 aspects listed under 5 marks 3 of 5 aspects listed under 5 marks 4 of the 5 aspects listed under 5 marks correct. Correct translation of 'exceeded' into a mathematical inequality along with relevant fully labelled diagram; correct transformation of probability; correct calculations and answer. d. 0 of the 4 aspects listed under 4 marks 1 of the 4 aspects listed under 4 marks 2 of the 4 aspects listed under 4 marks 3 of the 4 aspects listed under 4 marks Correct answer with relevant working supported using statistical tables and fully labelled diagram. Sentences Sentence answering the question included for one or fewer parts Sentence answering the question included for parts b., c. and d.