Final answer:
The sampling distribution of the sample proportion of Americans with hearing trouble, given a large sample size of 120 and both np and nq greater than 5, would be approximately normal according to the central limit theorem.
Step-by-step explanation:
When we are looking at the sampling distribution of the sample proportion in this case, we are essentially analyzing the distribution that would result from taking many samples of size 120 from the population and calculating the proportion of Americans with hearing trouble in each sample. Given that 15% of all Americans have hearing trouble, we can denote this population proportion as p = 0.15 and the complement, those without hearing trouble, as q = 0.85.
To determine the shape of the sampling distribution, we can apply the central limit theorem for proportions, which states that if the sample size is large enough, the sampling distribution of p' (the sample proportion) is approximately normal with mean μp' = p and standard deviation σp' = √(pq/n), where n is the sample size.
In this case, since n is 120, which is a sufficiently large sample size, and np and nq are both greater than 5 (120 * 0.15 = 18 and 120 * 0.85 = 102), the sampling distribution will be approximately normal according to the central limit theorem. Therefore, the correct answer is Option 1: Approximately normal.