The United Kingdom Forestry Commission reports that 43% of hectares of woodland area in the United Kingdom were certified as "sustainably managed" in 2016. Suppose an employee took a simple random sample of 400 hectares and found that 47% of the sampled hectares had that certification in 2016.

Identify your information:
- [tex]N = 400[/tex]
- [tex]P = 0.43[/tex]

Check Conditions:

1. 10% Condition:
- We assume 400 hectares is less than 10% of all hectares.

2. Large Counts Condition:
- Check if [tex]400 \times 0.43[/tex] and [tex]400 \times (1 - 0.43)[/tex] are both greater than 10.

Is this approximately normal?

When checking for a normal approximation to the binomial distribution in the given scenario, the 10% condition is assumed to be met given the sample size and large expected population. The Large Counts condition is satisfied since both the expected number of successes (172) and failures (228) are greater than 10.

Explanation:

The question involves determining if the proportion of sustainably managed woodland area sampled is approximately normal based on the conditions presented. The key elements to address are the 10% condition and the Large Counts condition for a binomial distribution in order to use normal approximation.

The 10% condition requires that the sample size be less than 10% of the population to ensure independence between the samples. Since we do not know the total number of hectares, this condition cannot be fully assessed, but it's reasonable to assume that 400 hectares would be less than 10% of all UK woodland hectares, which would be in the millions.

For the Large Counts condition, we calculate the expected number of successes and failures using the sample probability. The expected number of successes is $400 \times 0.43 = 172$ and the expected number of failures is $400 \times (1 - 0.43) = 228$. Both values need to be greater than 10 for the normal approximation to be valid, which they are. Therefore, the Large Counts condition is satisfied and the distribution of the sample proportion can be considered approximately normal.