The shape of the sampling distribution of p, the proportion of early decision applicants, will be approximately normal thanks to the Central Limit Theorem. Large samples (20 students) from any population tend towards a bell-curve shape, regardless of the original distribution. Therefore, expect values centered around 11% with most falling within a few percentage points above and below
Option C is correct.
The shape of the sampling distribution of p (the proportion of freshmen using the early decision option) will be approximately normal (C), thanks to the Central Limit Theorem (CLT).
Here's the reasoning:
The CLT states that for sampling from any population, regardless of its original distribution, the sampling distribution of the mean or proportion will be approximately normal as the sample size increases.
In this case, we're dealing with sample proportions (p) drawn from a population of 250 freshmen, where 11% used the early decision option. We're also taking relatively large samples (20 students).
Although the original population might not be perfectly normal, the CLT assures that for samples of size 20, the distribution of p will be close to normal with:
Mean: ~11% (the population proportion)
Standard deviation: [sqrt(p*(1-p))/sqrt(n)] ~ 4.5% (calculated using the formula)
Therefore, we can expect the sampling distribution of p to be centered around 11%, with most values falling within a few percentage points above and below this central point, resembling a bell curve.
While some deviations from perfect normality might exist, especially for extreme values of p, the overall shape will be predominantly normal due to the large sample size and the CLT's applicability.
So, the final answer is: C. Approximately normal.