To solve this problem, we will use the normal approximation to the binomial distribution.
First, we need to check that the conditions for using this approximation are met. The conditions are:
- The sample size is large enough: n p and n (1-p) are both greater than or equal to 10, where n is the sample size and p is the probability of success.
- The observations are independent.
For this problem, n = 540 and p = 0.06. Therefore, n p = 32.4 and n (1-p) = 507.6, which are both greater than 10. The observations are also assumed to be independent. Therefore, we can use the normal approximation to the binomial distribution.
Next, we need to calculate the mean and standard deviation of the sampling distribution of the proportion of flops in a sample of 540 films.
The mean of the sampling distribution is equal to the population proportion:
μ = p = 0.06
The standard deviation of the sampling distribution is given by:
σ = sqrt(p (1-p) / n)
σ = sqrt(0.06 * 0.94 / 540)
σ = 0.01823 (rounded to 5 decimal places)
We want to find the probability that the proportion of flops in a sample of 540 released films would be greater than 8%. We can standardize this value using the formula:
z = (x - μ) / σ
where x is the value we are interested in (0.08), μ is the mean of the sampling distribution (0.06), and σ is the standard deviation of the sampling distribution (0.01823).
z = (0.08 - 0.06) / 0.01823
z = 1.098
Using a standard normal distribution table or calculator, we can find the probability that z is greater than 1.098 to be approximately 0.1365. Therefore, the probability that the proportion of flops in a sample of 540 released films would be greater than 8% is approximately 0.1365, rounded to four decimal places.