1. A game is designed based on the use of a pair of dice and adding the scores shown when the dice are rolled. The game centres around throwing primes. Write the probability distribution for the five primes and group all the non-primes together. Identify the most likely result. [6]
The game costs D (a multiple of five money units) to join for each roll of the dice; this is called a ‘stake’. The rewards are given by the formula 5p where p is the prime, so the stake is spent, but a return is possible. Calculate the expected reward from the game and comment on the value of D that you think is reasonable. [6]
2. A business manufactures widgets. The business manager is concerned that the machine used to make the widgets, the Big Widget Maker, is not behaving as well as he would like. The BWM produces 1200 widgets an hour. Samples of ten widgets are taken every hour and tested (non-destructively) for quality.
After two 18-hour days of testing, the figures for failures are as follows:
2 1 0 2 0 2 1 0 0 1 0 1 1 1 0 1 2 1
2 1 0 3 0 3 0 1 0 0 1 0 2 1 1 0 1 0
Copy and complete the frequency table for this data: x 0 1 2 3 4 5 6
N 0 0 0
Use the table to calculate the mean and standard deviation of the data. [5]
Describing the variate you are using, compare the table of figures with a binomial distribution where p or q is 0.9 and comment as to whether you think this is a suitable distribution and whether some other value of probability would give a better fit. [6]
3. Candidates applying to join the Air Force are subjected to a complete battery of tests. The probability that a candidate passes a particular test, test A, is thought to be 1/3. Suppose that test B has a probability of success of 5/8 and that test C has a success rate of 3/5.
a) What is the probability of a candidate passing all three? [1]
b) What is the probability of a candidate failing exactly one test? [2]
c) What is the probability of at least 16 out of 20 students passing test B? (4 s.f. [4]
d) What is the probability of 6 students producing at least one successful candidate ? [3]
e) How many students do you need to test so as to have a 95% probability of at least one successful candidate? [5]
4. The mass of 15 small birds has been sampled and the data collected are as follows:
23 24 25 28 29 27 23 25 27 21 22 26 34 17 22
a) Find the mean and standard deviation of the data [5]
b) Establish some bounds for outliers [2]
c) The one apparently very large bird has been incorrectly identified and is a different breed of bird. Remove this item of data and recalculate the mean and standard deviation. [3]
d) Comment on outliers for the adjusted figures. [2]
DJS 20081106
12, 11, 15, 12 => 50 total
UNIVERSITY OF CAMBRIDGE UNINTENTIONAL EXAMINATIONS
General Certificate of Education
Advanced Subsidiary Level and Advanced Level
MATHEMATICS A2 9709/05
Paper 5 Sadistics 1 (S1) November 2008Up to 1.5 hours
Additional Materials: Answer Booklet/Paper
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your English Name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid (all of these are forbidden examination materials).
Make sure your phone is turned off. Make sure anything that could be described as notes is well out of reach.
Put any drink out of sight and leave it there. Check that you are wearing your uniform properly.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate. Formulae are assumed to be known.
You are reminded of the need for clear presentation in your answers.
At the end of the test, hand in ALL the papers you received; both question paper and your worked answers.
Make sure your name is on every answer sheet.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
This proved much too difficult for this year’s students - basically, they can’t read as well as they think.
Q1 five primes, distribution has six entries. Expectation should (be calculated and) just favour the owner of the game.
Q2 The distribution matches a binomial of B(36,0.91) well. No S2 techniques required. Few described the variate (can’t read, you see).
Q3 ‘at least’ ignored my most candidates. Last answer: 23 or more.
Q4 If the µ±2ø outluiers is used, the 17 becomes an outlier. Not so with the 1.5(IQR) - style definition.
