Upper Middle School


1.  Attendance to date this term at Supervised Prep is, after 15 nights, 16,17,15,12,16,13,9,15,13,9,9,10,14,15,8.

i)  Create an ordered stem and leaf diagram using groups of five; 5-9, 10-14 etc[4]

ii)  Calculate the mean and variance                                                                        [3]

iii)  Comment on the distribution including at least one other measure each of dispersion and location (or central tendency)                                                                                   [2]

iv)  Tonight’s attendance brings the mean up to 13. Find the new variance.            [3]

v)  The equivalent figures in Junior Prep for 15 nights are µ = 35 and s = 5.0. Based on your figures for 15 nights, what is the (older) attendance at Supervised Prep equivalent to only 24 juniors?                                                                                                                      [3]


2.  Neither Sean nor Siobhan can get to the library this week, so Mrs Lea goes on their behalf with their tickets (up to 3 books each). She knows they like Terry Pratchett – so does she – and decides that she will pick up to four from the eighteen she finds together

i)  In how many different ways can she do this?                                                                   [2]

ii)  On consideration, Mrs Lea is pretty sure she has already seen 5 of the 18 around the house. If she picked exactly 4 books at random, how many selections will include none of these five?
                                                                                                                                               [3]

iii)  When she has picked six books for the children and returned home, expecting them to read all of them (both read all six), she finds that they have had a disagreement and are staying in separate rooms. In how many ways can she give them three books each?                         [2]

iv)  What was the probability that, on the library shelf, the five books already read were adjacent to each other?                                                                                                                       [4]

v)  Sean has, in fact, read 11 of the 18 books his mother found. Explain why, given that she picked less than four Pratchetts eventually, that you can be certain he has read one of these already.                                                                                                                                  [4]


3.  Dean has been reviewing the options made by his French class before starting GCSE courses. Those particularly good at languages were permitted to add another (German or Spanish) to their list and also had to choose a Technology, of which one option was ICT. Half the class do a second language, splitting 3:2 in favour of German; few of these linguists also do IT – a fifth of the German linguists and a quarter of the Spanish. 60% of the others picked IT over the other technology options.

Find the probability that someone chosen at random

i)    does IT     ii)   does Spanish and IT                                                                            [2,2]

ii) does one of Spanish and IT  iv)  does Spanish, given that they do IT                           [2,3]

70% of those doing two languages at GCSE go on to do an A-level language. Stating any assumptions you make, how many of the 65 in the year would you expect to take a language at A-level? Can you give a probable range for this answer?                                                  [3,1]


4.  In a multiple choice Biology paper of twenty questions, each with five alternative answers, James has done too little revision and so guesses every answer. Testing at the 5% level; and explaining your thinking in each case.

i)  If you assume a binomial distribution applies, what range of marks can he expect?        [5]

ii  )Helen finds she can answer 5 questions with certainty but guesses the rest. What marks reflect this approach?                                                                                                             [3]

iii)  Alex expects to answer 5 correctly, to have a 50% chance with the next 5 and will guess the rest. Give Alex’s likely range of marks.                                                                                   [3]

iv)  James scores none on one of these tests.  Is he still guessing them all?                        [3]

v)  Later on, James scores nine. How many is he guessing now?                                         [3]


                            [60]


Sean, Siobhan, Steve, Simon, Sharon, Susan were all in the same family. Dean disppeared from my radar, as did James, who I saw in a car more than once, but never spoke to once he left school.

A pity; it is nice to know where pupils go.

© David Scoins 2017