Sports of Sorts

1. The results of one club’s 19 runners in the Plymouth Half Marathon this year were as follows, measuring to the nearest minute: 1:15. 1:23, 1:24, 1:28, 1:30, 1:32, 1:34, 1:34, 1:35, 1:36, 1:37, 1:37, 1:38, 1:38, 1:42, 1:43, 1:46, 1:46, 2:18.

(i)  Construct an ordered stem and leaf diagram grouped in 15 minute intervals        [2]

(ii)  Find the median and the inter-quartile range for this data                                    [4]

(iii)  Show that the mean is 1:36:38 [1 hour, 36 mins, 38 secs] and find the standard.deviation.   [4]

(iv)  By calculating the skewness, or otherwise, comment on the distribution.                      [1]

(v)  Establish where outliers would be for this data and confirm that only the slowest runner (who made the mistake of setting off with the leaders and thus not producing the expected sub 1:40 time) so qualifies, Recalculate the mean and s.d without this figure, ruling it as an error.      [4]

2.  Snooker is a game played in turn at a table. Each match is made up of frames. There are 15 red balls of value one point and six different colours of value 2 to 7. In a turn a player will attempt to “pot” balls, alternating each red with any colour until failing to make a pot, at which point the other player has a turn. The score of a turn is called a “break”. When all the reds have been potted (and the subsequently chosen colour played and returned to the table), the colours are potted in ascending order of value.

Snooker players can be modelled as having a consistent probability p of potting the next ball, player A at pA and so on; it is assumed that shots are independent and that this modelling can be applied to every shot taken.

(i)  Player A is a novice and pA = 0.1. Draw a tree diagram showing a turn at the table of up to 4 strokes [attempts at pots] and thus find the likelihood of going further.                                 [5]

(ii)  Player B is by camparison an expert and reckons to score century breaks about one frame in 50. Show that this is consistent with potting sequences of 24 to 36 balls                         [4]

(iii)  Calculate a range of values for pB .                                                                                 [5]

3.  The House needs to pick a soccer team from the GCSE years. Find the number of ways in which the eleven can be selected as each piece of information is added:

(i)  There are 20 year 11s and 12 year 10s                                                                              [2]

(ii)  The rules require exactly 5 year 10s to play                                                                      [2]

(iii)  Of the girls, only Jo in Year 11 is prepared to play.
       There are 13 Yr 11 boys and 8 in Yr10                                                                            [3]

(iv)  One boy in each year (doesn’t play and) won’t play at all; two in each year are only prepared to play in goal, if they play at all. Jo discovers that all the proposed games clash with her County hockey games.                                                                                                             [6]

(v)  Graham, Chris and Michael of year 11 all play for the same good club side.                    [2]

4.  A football-loving lager lout wanders into an Amsterdam bar while still relatively sober. He is delighted to find 16 gorgeous babes lined up at the bar. Little does he realise that the probability that any one of these is a cross-dressing male is a third. Now answer the following unlikely questions:

(i)  What is the expected number and the most likely number of girls at the bar?             [2]

(ii)  If he were to pick three at random, what is the probability that

A)  exactly one is a girl               B)    at least one is a girl                                                  [3]

(iii)  He is entirely unaware of the gender situation and when warned by a firend that bars like this exist here, denies that this is a case in point. Carefully write a hypothesis test to establish the critical region at the 5% significance level.                                                                  [6]

(iv)  How does the test and its result change once he believes that any of those at the bar are blokes? What assumptions are being made for the test to be valid?                                [4]

© David Scoins 2017