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 calculate $P(X > 6)$ using the binomial model $X \sim B(8, 0.76)$:

We need to find the probabilities for $P(X = 7)$ and $P(X = 8)$ and sum them:

$P(X > 6) = P(X = 7) + P(X = 8).$

Using the binomial probability formula:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k},$

where $n = 8$ and $p = 0.76$,

1. For $P(X = 7)$: $P(X = 7) = \binom{8}{7} (0.76)^7 (0.24)^1 \approx 0.27.$

2. For $P(X = 8)$: $P(X = 8) = \binom{8}{8} (0.76)^8 (0.24)^0 \approx 0.20.$

Thus,

$P(X > 6) \approx 0.27 + 0.20 = 0.47.$

To determine the expected number of days with a daily mean total cloud cover of 7, we implement the formula for expectation in a binomial distribution:

Expected days = $n \cdot p = 184 \cdot P(X = 7)$ where $P(X = 7)$ is calculated from previous steps.

Using the value calculated above ($P(X = 7) \approx 0.27$):

$Expected = 184 \cdot 0.27 \approx 49.68 \approx 50 (to 1 decimal place).$

The results from part (b) suggest that the expected number of days with a cloud cover of 7 (approximately 50) closely aligns with the data obtained (52 days). This similarity indicates that the model fits reasonably well but must be analyzed further, as factors affecting cloud cover may lead to discrepancies. Therefore, while there is initial support for using the binomial model, additional scrutiny is required to evaluate its overall suitability.

To find the proportion of days when the daily mean total cloud cover was 6 or greater, we can assess the counts of days: There are 28 days noted, and to find how many of these are 6 or greater, we see that:

• Days with 6 or greater = 6 (1) + 7 (2) + 8 (3) = 6.

Therefore, the required proportion is:

$\frac{6}{28} \approx 0.214.$

This indicates around 21.4% of the days had a cloud cover of 6 or greater.

Considering the proportion of days with a cloud cover of 6 or greater (approximately 21.4%), it raises questions about the validity of Magali’s binomial model. The model suggests a greater probability of high cloud cover, while the actual data presents a lower occurrence of high cloud cover days, therefore indicating that Magali’s model may not adequately represent the observed daily cloud cover data.