According to a 2017 Pew Research report, 40 % of millennials...

According to a 2017 Pew Research report, $40 \%$ of millennials have a BA degree. Suppose we take a random sample of 500 millennials and find the proportion who have a driver's license. Find the probability that at most $35 \%$ of the sample has a BA degree. Begin by verifying that the conditions for the Central Limit Theorem for Sample Proportions have been met.

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Key Concepts

Central Limit Theorem for Sample Proportions

This theorem states that when a sample is sufficiently large, the distribution of the sample proportion will be approximately normal, regardless of the shape of the population distribution. This property allows us to make probability statements about the sample proportion using the normal distribution, even if the actual underlying distribution is binomial.

Conditions for Applying the Central Limit Theorem

Before applying the CLT to a sample proportion, several conditions must be checked: the sample must be randomly selected to ensure independence; the independence or 10% condition must be satisfied, meaning the sample size is less than 10% of the population if sampling without replacement; and the success-failure condition must hold, which requires that both the expected number of successes (np) and the expected number of failures (n(1-p)) are at least 10.

Sample Proportion

The sample proportion is the fraction of individuals in the sample who exhibit a particular characteristic. It is a random variable representing the estimate of the true population proportion and serves as the basis for constructing a sampling distribution that, under appropriate conditions, approximates a normal distribution.

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