June 2018 Paper 3 Q1

EdexcelCurrent spec5 marksDRVs Large Data Set

1. Helen believes that the random variable \(C\), representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.

(a) Write down the probability distribution for \(C\). (2)

(b) Using this model, find the probability that cloud cover is less than 50% (1)

Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than 50% was 0.315

(d) Suggest an appropriate refinement to Helen’s model. (1)

Mark scheme (a)

Scheme

Marks

\(c\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)
\(\mathrm{P}(C = c)\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)	\(\frac{1}{9}\)

B1
B1ft

1.2
1.2

(2)

Notes

1^st B1 for a correct set of values for \(c\). Allow \(\left\{\frac{1}{8}, \frac{2}{8}, \ldots \frac{8}{8}\right\}\)

2^nd B1ft for correct probs from their values for \(c\), consistent with discrete uniform distrib’n
Maybe as a prob. function. Allow \(\mathrm{P}(X = x) = \frac{1}{9}\) for \(0 \leqslant x \leqslant 8\) provided \(x = \{0, 1, 2, \ldots, 8\}\) is clearly defined somewhere.

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{P}(C \lt 4) = \frac{4}{9}\) (accept 0.444 or better)	B1	3.4
	(1)

Notes

B1 for using correct model to get \(\frac{4}{9}\) (o.e.)

SC Sample space \(\{1, \ldots, 8\}\) If scored B0B1 in (a) for this allow \(\mathrm{P}(C \lt 4) = \frac{3}{8}\) to score B1 in (b)

Mark scheme (c)

Scheme	Marks	AO
Probability lower than expected suggests model is not good	B1ft	3.5a
	(1)

Notes

B1ft for comment that states that the model proposed is or is not a good one based on their model in part (a) and their probability in (b)

\(|\text{(b)} - 0.315| \gt 0.05\) Allow e.g. “it is not suitable”; “it is not accurate” etc
\(|\text{(b)} - 0.315| \leqslant 0.05\) Allow a comment that suggests it is suitable
No prob in (b) Allow a comparison that mentions 50% or 0.5 and rejects the model
No prob in (b) and no 50% or 0.5 or (b) \(\gt\) 1 scores B0
Ignore any comments about location or weather patterns.

Mark scheme (d)

Scheme	Marks	AO
e.g. Cloud cover will vary from month to month and place to place So e.g. use a non-uniform distribution	B1	3.5c
	(1)
	(5 marks)

Notes

B1 for a sensible refinement considering variations in month or location
Just saying “not uniform” is B0

Context & “non-uniform” Allow mention of different locations, months and non-uniform or use more locations to form a new distribution with probabilities based on frequencies
Context & “binomial” Allow mention of different locations, months and binomial
Just refined model Model must be outlined and discrete and non-uniform e.g. higher probabilities for more cloud cover or lower probabilities for less cloud cover
Continuous model Any model that is based on a continuous distribution. e.g. normal is B0