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

$P(X > 6)$ can be found using the complementary rule:

$P(X > 6) = 1 - P(X \leq 6) = 1 - P(X = 0) - P(X = 1) - P(X = 2) - P(X = 3) - P(X = 4) - P(X = 5) - P(X = 6)$

Using the frequencies:

$P(X \leq 6) = \frac{0 + 1 + 4 + 4 + 7 + 10 + 52}{184} = \frac{78}{184}$

Thus,

$P(X > 6) = 1 - \frac{78}{184} = \frac{106}{184} = 0.5761$

To find the expected number of days with a daily mean total cloud cover of 7, we can use the binomial model with parameters (n = 184, p = 0.76).

The expected value (mean) of the binomial distribution is given by:

$E(X) = n \cdot p = 184 \cdot P(X = 7)$

From the probability previously computed, we can find the estimated number:

$E(X) = 184 \cdot 0.2811 \approx 51.735\text{ days}$

Rounded to 1 decimal place, the expected number of days is 51.7.

The results from part (b) suggest a comparison between actual outcomes and model predictions. The computed probabilities should ideally align with what is observed in the data. Since the probability found in part (b)(i) indicates a notable difference between predicted and observed distribution of cloud cover days, it raises concerns about the suitability of Magali's binomial model.

If the actual distribution demonstrates much higher occurrences of lower cloud cover days or deviates significantly from the model's expectation of higher means, this indicates that the model cannot adequately capture the observed phenomena.

The proportion of days when the daily mean cloud cover was 6 or greater among the 28 days can be calculated:

From the subsequent table, we have:

• Days with cloud cover of 6: 6 days

Thus, the proportion is:

$P(X \geq 6 | 28) = \frac{6}{28} = 0.2143$

The observed proportion of days with high daily mean cloud cover (greater than or equal to 6) derived from the dependent condition suggests there is a greater likelihood of observing high cloud cover, contradicting Magali's initial binomial distribution model.

This indicates that the binomial model may not be suitable as it does not accurately reflect the cloud cover trends or dependencies observed in the data.