Helen believes that the random variable C, representing cloud cover from the large data set, can be modelled by a discrete uniform distribution - Edexcel - A-Level Maths Statistics - Question 1 - 2018 - Paper 2

For the discrete values considered (0 to 8), the values less than 50% correspond to C = 0, 1, 2, 3 or 4. Thus, we calculate:

$P(C < 4) = P(C = 0) + P(C = 1) + P(C = 2) + P(C = 3) + P(C = 4)$

Since all probabilities are equal:

$P(C < 4) = 5 \times \frac{1}{9} = \frac{5}{9} \approx 0.555$.

Therefore, the probability that cloud cover is less than 50% is approximately 0.555 or 55.5%.

Helen’s model may not be suitable based on the observed data from 2015, which indicated that only 31.5% of days had cloud cover of less than 50%. Hence, the expected probability of 55.5% as per the model significantly differs from the empirical data. The variation suggests that the cloud cover is not uniformly distributed across days, leading to the conclusion that the discrete uniform distribution does not accurately represent the real data.

A suitable refinement to Helen’s model could be to use a non-uniform distribution that accounts for variations in cloud cover depending on time and location. This might involve applying a model that reflects seasonal changes in cloud cover or environmental factors influencing weather patterns, potentially using a continuous distribution instead of a discrete one.