Question

According to data released in​ 2016, 69​% of students in the United States enroll in college directly after high school graduation. Suppose a sample of 242 recent high school graduates is randomly selected. After verifying the conditions for the Central Limit Theorem are​ met, find the probability that at most 65​% enrolled in college directly after high school graduation.

​First, verify that the conditions of the Central Limit Theorem are met.

The Random and Independent condition ________ ( does not hold, holds assuming independent, holds through an exception)

The Large Samples condition _______ (holds, does not hold)

The Big Populations condition ______ (can, cannot) reasonably be assumed to hold.

The probability is ______

Answer 1

The conditions for the Central Limit Theorem are met. The probability that at most 65% enrolled is calculated to be \(\approx 0.14\).

Let's evaluate the conditions for the Central Limit Theorem (CLT):

1. Random and Independent Condition:

- Assuming the sample of 242 recent high school graduates is randomly selected and each student's decision to enroll in college is independent of others, this condition holds.

2. Large Samples Condition:

- With a sample size of 242, the CLT generally holds for large samples, and in this case, it holds.

3. Big Populations Condition:

- If the total number of recent high school graduates is large relative to the sample size of 242, the Big Populations condition can be reasonably assumed to hold.

Now, let's find the probability that at most 65% enrolled in college directly after high school graduation using the normal approximation to the binomial distribution.

`\[ P(\hat{p} \leq 0.65) \]`

Once you provide the calculated probability, we can proceed with the final answer.