The probability that between 79% and 81% of the 500 sampled wildlife watchers actively observed mammals in 2011 is approximately a) 0.42.
1. Central Limit Theorem: Since we are taking a random sample of 500 individuals from a large population (over 71 million), the Central Limit Theorem applies. This means that even if the population proportion is not normally distributed, the sample proportion will be approximately normally distributed, regardless of the population distribution, as long as the sample size is sufficiently large.
2. Calculating Standard Error: The standard error of the proportion (SE) represents the typical variability of sample proportions from the population proportion. We can calculate it using the formula:
SE = sqrt(p * (1 - p) / n)
where:
* p = population proportion (0.8)
* n = sample size (500)
SE = sqrt(0.8 * (1 - 0.8) / 500) ≈ 0.016
3. Calculating Z-scores: We want to find the probability of the sample proportion falling between 79% and 81%. We can convert these percentages to z-scores using the formula:
z = (sample proportion - population proportion) / SE
* Lower bound (79%): z = (0.79 - 0.8) / 0.016 ≈ -0.63
* Upper bound (81%): z = (0.81 - 0.8) / 0.016 ≈ 0.63
4. Finding the Probability: We need to find the area between -0.63 and 0.63 under the standard normal curve. Using a standard normal distribution table or calculator, we can find this area to be approximately 0.423.
Therefore, the approximate probability that between 79% and 81% of the 500 sampled wildlife watchers actively observed mammals in 2011 is 0.42, which corresponds to option a).