Final answer:
For a sample proportion to be approximately normally distributed, both np and n(1-p) must be greater than 10. In case (a), at least 17 more respondents are needed; while in case (b), no additional respondents are necessary, but more would be preferable for accuracy.
Step-by-step explanation:
The question revolves around the concept of sampling distributions and the conditions required for those distributions to be approximately normal when estimating population proportions. These conditions are often related to the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough.
In cases (a) and (b), to say that the distribution of the sample proportion of adults who respond 'yes' is approximately normal, we use the rule of thumb that the sample size (n) should be such that both np and n(1-p) are greater than 10, where p is the proportion of successes in the population. This requirement helps ensure the normal approximation can be applied.
(a) If 15% of all adult Americans support the proposed changes, the calculations would be as follows: np = 50 * 0.15 = 7.5 and n(1-p) = 50 * 0.85 = 42.5. Since np is not greater than 10, we need to increase our sample size. If n is the new sample size, then we need n * 0.15 > 10, which gives n > 66.67. Since the researcher already has a sample of 50, he/she needs at least 17 more adult Americans (rounding up).
(b) Similarly, for 20% support, we calculate np = 50 * 0.20 = 10 and n(1-p) = 50 * 0.80 = 40. In this case, np is exactly 10, so technically the sample size is barely adequate, but to ensure a more accurate normal approximation, sampling more than 50 individuals would be preferable. However, since np already meets the threshold, no additional samples are strictly necessary for this case based on the rule of thumb.