Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set - Edexcel - A-Level Maths Statistics - Question 4 - 2019 - Paper 1

Using the binomial distribution, we find:

$P(X > 6) = 1 - P(X \leq 6)$ To find $P(X \leq 6)$, we calculate:

$P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)$ Using the binomial formula, where $n=8$ and $p=0.76$:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ Calculating these probabilities gives:

$P(X > 6) \approx 0.703$

The expected number of days with a total cloud cover of 7 can be calculated using:

$E[X=7] = n \cdot p$ With $n = 184$ and $p = P(X=7)$; From earlier calculations, we can estimate $P(X=7) \approx 0.281$. Thus,

$E[X=7] = 184 \cdot 0.281 \approx 51.784\approx 51.8$

The probabilities obtained in part (b) show a significant difference between $P(X \geq 6)$ (approximately 0.4348) and the expected $P(X=7)$ (approximately 0.281). This indicates that there is a greater likelihood of days with a higher cloud cover than predicted by the model, suggesting that Magali's binomial model may not be suitable.

Out of the 28 days, since all were those with a daily mean total cloud cover of 8, we can say:

$\text{Proportion} = \frac{28}{28} = 1.0$ Thus, 100% of these days had a cloud cover of 6 or greater.

Given that all sampled days had a cloud cover of 8, indicating a higher proportion of high cloud cover, this suggests that the model may not accurately reflect the observed data. Therefore, Magali's model, which assumes a binomial distribution, may not be suitable as it does not account for the observed frequency of higher cloud covers.