Lower Sixth

1.The 1995 class have completed six practice papers, with their results displayed below

Class range   0-9   10+   15+  20+  25+  30+ 35+ 40+ 45-49  
frequency       0        6      9      7     12      9    18     7        12

(i)  Find the median and inter-quartile range to the nearest half mark                                        [3]

(ii) Calculate the mean µ and standard deviation, s                                                                   [3]

(iii)  Only ten attended the last practice paper:  they produce a mean of 37.9 and a variance s² of 69.09. Find the new aggregate µ and s for these 90  marks                                                   [3]

(iv) This year the board will create xm, the exam module mark, according to the formula 12xm = 56/50 µ + 9,   so you can now calculate the µ and s for the exam module marks for the ten (µ = 37.9, s = 69.09) who sat the last paper.                                                                                    [3]

2.   Professor Kurti, celebrated Oxford physicist and cook, has the delightful habit of using an alcohol-filled syringe to enhance his sweets. Duncan has made a dozen mince pies to try on four friends coming to tea (among other foods) and with access to the drinks cabinet has made them all different. He particularly likes (hint of original question) the eight fruit liqueurs.

i)  How many selections are available to Duncan after he’s passed the plate around once?  [2]
ii)  Show that 168 of these leave six of the fruit liqueurs on the plate                                     [3]
iii)  Construct a table showing the complete distribution, ie all 8, just 7,.., just 5…                [4]
iv)  Which outcome is the most likely?                                                                                    [1]
v)  What is the expected outcome?                                                                                         [2]

3.   This year a large proportion of the Upper Sixth sat general Studies in the M-K Hall in 9 columns and 11 rows, seated in alphabetic order of surname, as is usual. Of the candidates that year, the 15 girls were randomly distributed, partly because the first letter of surname is independent of gender. In order to use statistical tables, where appropriate take the number of candidates to be 100. Find the probability that

i)  a desk chosen at random is currently used by a boy                                                    [ 1]
ii)  a column includes only one girl                                                                                     [2]
iii)  a column includes two girls                                                                                          [2]
iv)  a column includes exactly two girls at adjacent desks                                                [2]
v)  I find I can identify only 65 of the 100 with certainty and count how many of these sit in any column. What range of values would you expect?                                                            [3]
vi)  I know everyone in the back row. What is the greatest number in any column that I can reasonably expect to find in ‘adjacent’ seats [in that column]?                                         [3]

4.  For candidates aiming at A grades, their marks, M, on Question 4 of S1 quite clearly match the binomial distribution curve M ~ Bin(13,0.9), i.e. the probability of any mark is 0.9 and each mark is independent of the others. [In 1995 there were 13 marks per question].

Working at the 5% confidence level, find the probability of:
i)  full marks     ii)  better than ten     iii) ten or better                                                        [1,3,1]
iv)  Of eight practice papers, Sophie scores less than ten only once. Test the theory that she can expect this only rarely. Explain in detail.                                                                               [4]
v)  Duncan has scored once less than twelve. Using the same basic argument, test whether he can expect less than twelve only rarely.                                                                                [2]
vi)  Can you produce an expression which will calculate what size sample has one exception (as above) and classifies as unusual?                                                                                         [2]

Duncan read Law (with German) at Oxford, Sophie read Medicine at UCL – and both managed an A in Maths (and in S1)                       [12, 12, 13, 13] => 50 in all

© David Scoins 2017