According to data released in 2016 , 69 % of students in the...

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 200 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. (Source: nces.ed.gov)

Key Concepts

Central Limit Theorem

This theorem states that the sampling distribution of a statistic (such as the sample mean or proportion) will approximate a normal distribution, provided the sample size is sufficiently large. This is key when making inferential statements because it allows us to use normal probability methods even if the original data is not normally distributed.

Sampling Distribution of a Proportion

When dealing with proportions, the sampling distribution is the distribution of sample proportions taken from all possible samples of a given size. Its mean is the true population proportion, and its variability is measured by the standard error, which depends on both the population proportion and the sample size.

Standard Error

The standard error quantifies the variability of a sample statistic from sample to sample. For a sample proportion, it is calculated based on the population proportion and the sample size, and it is essential for assessing how much a sample proportion might differ from the true population proportion.

Normal Approximation and Z-Score Standardization

Once the sampling distribution is established as approximately normal (through the Central Limit Theorem), the z-score is used to standardize a sample proportion. This standardization transforms the sample statistic into a value that can be compared to a standard normal distribution, thereby facilitating the calculation of probabilities associated with the sample proportion.

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