THIS ASSESSMENT NOTIFICATION CAN BE FOUND ON THE COLLEGE WEBSITE UNDER THE LEARNING AND TEACHING TAB

Transcription

1 Student Name: Teacher: Assessment Task for Stage 6: HSC Subject: General Mathematics THIS ASSESSMENT NOTIFICATION CAN BE FOUND ON THE COLLEGE WEBSITE UNDER THE LEARNING AND TEACHING TAB Assessment Task No. 3 Date: Thursday 18 th June 2015 Weighting 20 % Submission Instructions This is an in-class task which will be completed in the Hall, period 4 for the duration of 55 minutes Penalty for non attendance On the date of the assessment will immediately receive a 50% mark penalty of the achieved mark pending Illness/Misadventure certification. On the day following the assessment will receive a zero mark pending Illness/Misadventure certification. Outcomes being assessed MG2H-1 uses mathematics and statistics to evaluate and construct arguments in a range of familiar and unfamiliar contexts MG2H-2 analyses representations of data in order to make inferences, predictions and conclusions MG2H-3 makes predictions about situations based on mathematical models, including those involving cubic, hyperbolic or exponential functions MG2H-6 makes informed decisions about financial situations, including annuities and loan repayments MG2H-7 answers questions requiring statistical processes, including the use of the normal distribution, and the correlation of bivariate data MG2H-8 solves problems involving counting techniques, multistage events and expectation MG2H-9 chooses and uses appropriate technology to locate and organise information from a range of contexts MG2H-10 uses mathematical argument and reasoning to evaluate conclusions drawn from other sources, communicating a position clearly to others, and justifies a response. Outcomes omitted: MG2H-4 and MG2H-5 Outcomes added: MG2H-3, MG2H-6 and MG2H-7

2 1. Description of the Task TASK DETAILS Students will receive a sample of the questions (with answers) required to complete the task. The student is asked to. Answer similar questions to those on the sample paper in a class task of 55 minutes duration on Thursday 18 th June. The task will be divided into three sections covering topics: Multi-Stage Events, Annuities and Loan Repayments and Normal Distribution. Bring your own pens, pencils, Board of Studies approved calculator, ruler and eraser. There is to be no borrowing of equipment during the test. Attempt all questions on the test paper. Allocated marks will be shown on the paper. Show all necessary working. Marks will be deducted for careless or badly arranged work. Hand up the paper in three sections - Section A, Section B and Section C. Subject Specific Terminology: Multi-Stage Events: arrangement, expected number, expected value, experimental result, financial expectation, financial gain, financial loss, multiplication principle, multistage event, ordered selection, probability tree diagram, sample space, simulation, tree diagram, trial, two-stage event, unordered selection. Annuities and Loan Repayments: annuity, annuity stream, future value of an annuity, initial amount, interest, interest factor, loan repayment table, periodical contribution, present value of an annuity, principal, repayment Normal Distribution: bell-shaped, data set, normal distribution, population characteristic, population mean, population standard deviation, sample mean, standard deviation, standardised score, z-score 2. Classroom Learning: Students will be prepared to effectively complete this task through: Learning to (skills) multiply the number of choices at each stage to determine the number of outcomes for a multistage event. establish the number of ordered and unordered selections that can be made from a group of different items. use the formula for the probability of an event to calculate the probability that a particular selection will occur. determine the number of ways in which n different items can be arranged use probability tree diagrams to solve problems involving two-stage events. calculate the expected number of times a particular event would occur, given the number of trials of a simple experiment. calculate expected value by multiplying each outcome by its probability and adding the results together. recognise that an annuity is a financial plan involving periodical, equal contributions to an account. calculate the future value of an annuity (FVA) using a table.

3 calculate the present value of an annuity (PVA) using a table. calculate the contribution per period, using a table. use a table to calculate loan instalments. calculate the total amount paid over the term of a loan. calculate the monthly repayment for a home loan from a table, given the principal, rate and term. describe the z-score corresponding to a particular score (in a set) as a number indicating the position of that score relative to the mean. use the formula to calculate z-scores. compare calculated z-scores from different data sets. use z-scores to make judgments in individual cases. identify and use properties of data that are normally distributed. Learning about (knowledge) multiplying the number of choices at each stage to determine the number of outcomes for a multistage event establishing the number of ordered and unordered selections that can be made from a group of different items using the formula for the probability of an event to calculate the probability that a particular selection will occur determining the number of ways in which n different items can be arranged using probability tree diagrams to solve problems involving two-stage events calculating the expected number of times a particular event would occur, given the number of trials of a simple experiment calculating expected value by multiplying each outcome by its probability and adding the results together recognising that an annuity is a financial plan involving periodical, equal contributions to an account. calculating the future value of an annuity (FVA) using a table. calculating the present value of an annuity (PVA) using a table. calculating the contribution per period, using a table. using a table to calculate loan instalments. calculating the total amount paid over the term of a loan. calculating the monthly repayment for a home loan from a table, given the principal, rate and term. describing the z-score corresponding to a particular score (in a set) as a number indicating the position of that score relative to the mean using the formula to calculate z-scores, comparing calculated z-scores from different data sets using z-scores to make judgments in individual cases. identifying and using properties of data that are normally distributed, Terms used in the assessing of this task: calculate, determine, evaluate, justify, list 3. Marking Criteria The marking criteria will be provided on the test paper.

4 HSC General Mathematics 2015 Sample Questions Multi-Stage Events Sample Questions 1) Six people A, B, C, D, E and F turn up to play in a tennis competition. a) List all possible ways two people can be chosen to play. b) What is the probability that E plays in the first game? c) What is the probability that A and F play first ANSWER: a) AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF. b) 1/3 c) 1/15 2) Phillipa s chances of passing a science test is 0.85 and of passing a history test is 0.7. Complete the tree diagram below to show the possible outcomes. 0.7 Pass Find the probability that she: 0.85 Pass Fail a) fails both tests. b) passes at least one of the tests. Pass Fail Fail ANSWER: a) b) ) A class is made up of 10 runners and 15 swimmers. If 2 students are chosen at random, what is the probability that both are swimmers? ANSWER: 7/20 4) A school committee has 6 students, 6 parents and 8 teachers. If two are selected at random, find the probability that neither are teachers. ANSWER: 33/95 5) Bianca plays a game by throwing three coins. The rules of the game are as follows. Bianca wins $60 if there are three tails. Bianca wins $15 if there are two tails. Bianca loses $30 if there is one or no tails. She plays the game 120 times. a) How many times would she expect to win $60? b) How many times would she expect to lose $30? c) What is the financial expectation of this game? Answer to the nearest cent. ANSWER: a) 15 b) 60 c) Loss of $1.88

5 6) Six girls and seven boys are attending a camp. a) A group of 4 girls is needed to cut firewood. How many groups can be chosen? b) A group of ten consisting of five boys and five girls is going canoeing. How many different groups can be chosen? ANSWER: a) 15 b) 126 Annuities Sample Questions 1. The following table shows the future value of an annuity with a contribution of $1. Future value of $1 Period 2% 4% 6% 8% 10% Use the table to calculate the future value of the following annuities: a) $3500 per year for 3 years at 10% p.a. compounded annually. b) $2000 per half-year for 2 years at 12% p.a. compounded biannually. ANSWERS: a) $ b) $ Joyce is saving to buy a boat. She deposits $2100 into an account at the end of every year for 4 years. The account pays interest of 8% p.a. compounding annually. Use the table from question 1 to calculate: a) the value of Joyce s investment at the end of 4 years. b) the interest Joyce earns on her investment over the 4 years. ANSWERS: a) $9 471 b) $1 071

6 3. The following table shows the present value of an annuity with a contribution of $1. Present value of $1 Period 1% 2% 4% 6% 8% Use the table to calculate the present value of $650 per quarter for 1 year at 8% p.a. compounded quarterly. ANSWER: $ The following table shows the monthly payments for each $1000 borrowed. Interest rate Period of loan 10 years 15 years 20 years 6 % p.a. $11.10 $8.44 $7.10 7% p.a. $11.61 $9.00 $7.75 8% p.a. $12.13 $9.56 $8.36 Billy is planning to borrow $ to buy a house at 7% p.a. over a period of 15 years. a) What is Billy s monthly payment on this loan? b) How much would Billy pay in total to repay the loan? c) How much would Billy save if he repaid the loan over 10 years? ANSWERS: a) $2 250 b) $ c) $ Normal Distribution Sample Questions 1. Which of the following frequency histograms shows data that could be normally distributed? ANSWER: A

7 2. The marks in a class test are normally distributed. The mean is 100 and the standard deviation is 10. a) Jason s mark is 115. What is his z-score? b) Mary has a z-score of 0. What mark did she achieve in the test? c) What percentages of marks lie between 80 and 110? You may assume the following: 68% of marks have z-scores between 1 and 1 95% of marks have z-scores between 2 and % of marks have z-scores between 3 and 3. ANSWER: a) z = 1.5 b) a score of 100 c) 81.5% of marks lie between 80 and A machine produces nails. When the machine is set correctly, the length of the nails are normally distributed with a mean of cm and a standard deviation of cm. To confirm the setting of the machine, three nails are randomly selected. In one sample the lengths are 5.950, and The setting of the machine needs to be checked when the lengths of two or more nails in a sample lie more than 1 standard deviation from the mean. Does the setting on the machine need to be checked? Justify your answer with suitable calculations. ANSWER: Suitable range = mean ± 1 standard deviation = to Lengths and are outside the range. Therefore the setting for the machine needs to be checked as 2 lengths in the sample are outside the range. 4. Two brands of light bulbs are being compared. For each brand, the life of the light bulbs is normally distributed. a) One of the Brand B light bulbs has a life of 400 hours. What is the z-score of the life of this light bulb? Brand A Brand B Mean b) A light bulb is considered defective if it lasts less than 400 hours. The following claim is made: Brand A light bulbs are more likely to be defective than Brand B light bulbs. Standard deviation Is this claim correct? Justify your answer, with reference to z-scores or standard deviations or the normal distribution. ANSWER: a) z-score is 2 b) For Brand A: z-score is 2. Therefore Both Brands are equally likely as 2.5% of both brands are likely to be defective.