Answer:
Hi,
Explanation:
To determine whether it's appropriate to use the normal distribution for probabilities in this scenario and apply the Central Limit Theorem, you need to check if the assumptions are satisfied. The Central Limit Theorem states that, for a random sample of a sufficiently large size, the sampling distribution of the sample proportion becomes approximately normally distributed, regardless of the shape of the population distribution, as long as certain conditions are met.
The conditions for using the Central Limit Theorem are:
Random Sampling: The samples must be selected randomly from the population to ensure that they are representative.
Independence: Each sample observation should be independent of the others. In the case of proportions, this means that the sample size should be small relative to the population size (n < 0.05N, where N is the population size).
Sample Size: The sample size should be "sufficiently large." The precise definition of "sufficiently large" can vary, but a commonly used rule of thumb is that the sample size (n) should be at least 30. In your case, n = 111, which is well above 30.
Success-Failure Condition: This condition requires that both np and n(1-p) should be greater than or equal to 5. In other words, you need a sufficient number of successes and failures in your sample. In this case, you have p = 0.59, so np = 111 * 0.59 ≈ 65.49 and n(1-p) = 111 * (1-0.59) ≈ 45.51. Both of these values are greater than 5, which satisfies the condition.
Given that your sample size is sufficiently large (n = 111) and the success-failure condition is met, you can assume that the sampling distribution of the sample proportion is approximately normal. Therefore, you can use the normal distribution to find the indicated probability.