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

To find $P(X > 6)$, we can calculate it as the complement of $P(X \leq 6)$:

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

We need the probabilities for each of these values:

• For $X=0$: 0
• For $X=1$: 1/184
• For $X=2$: 4/184
• For $X=3$: 7/184
• For $X=4$: 10/184
• For $X=5$: 52/184
• For $X=6$: 0

Calculating this gives:

$P(X \leq 6) = \frac{1 + 4 + 7 + 10 + 52}{184} = \frac{74}{184} = \frac{37}{92}$

Thus,

To find the expected number of days with a daily mean total cloud cover of 7,

Use the binomial distribution where:

• Number of trials (n): 184
• Probability of success (p): Frequency of cloud cover of 7, which is rac{28}{184} = 0.15217391.

So, $E(X) = n \times p = 184 \times 0.15217391 \approx 28.0\text{ days}$

Thus, the expected number of days with a daily mean total cloud cover of 7 is approximately 28, rounded to 1 decimal place.

The answers to part (b) show a discrepancy that may not support the use of Magali's model. The expected number of days with a daily mean total cloud cover of 7 is 28, while the actual frequency in the data set shows 52 days. This suggests that the binomial model's assumption may not accurately reflect the distribution of cloud cover, as the observed frequency for 7 significantly exceeds the expected count considering a model approaching 0.76 probability. Therefore, Magali's binomial model may not be suitable.

Calculating the proportion of days with a daily mean total cloud cover of 6 or greater out of the 28 days is necessary. We previously established that:

• Count of days with cloud cover of 6: 0 (based on the following day distribution)
• Count of days with cloud cover of 7 and 8: 2 + 1 + 1 = 4

Thus, $\text{Proportion} = \frac{4}{28} = 0.1428571\text{(or approximately 0.143)}$

This proportion reflects the relative frequency of higher cloud cover following days, indicating the rarity of such occurrences.

In light of the proportion calculated in part (d), which is approximately 0.143 for days with a daily mean total cloud cover of 6 or greater, we see a concerning mismatch with the assumptions of Magali's binomial model. The model expected a substantial proportion of higher cloud covers, yet the observed data diverges, displaying lower occurrences than predicted. Thus, it raises doubts about the suitability of Magali's model, as it does not appear to capture the actual patterns and trends in the data adequately.