According to a 2017 article in The Washington Post, 72 % of...

According to a 2017 article in The Washington Post, $72 \%$ of high school seniors have a driver's license. Suppose we take a random sample of 100 high school seniors and find the proportion who have a driver's license. Find the probability that more than $75 \%$ of the sample has a driver's license. Begin by verifying that the conditions for the Central Limit Theorem for Sample Proportions have been met.

Key Concepts

Central Limit Theorem for Sample Proportions

This theorem states that for sufficiently large sample sizes, the distribution of the sample proportion will be approximately normal, regardless of the shape of the population distribution. It underpins many statistical procedures that involve proportions, enabling the use of normal probability calculations even when the underlying data are binary.

Conditions for Normal Approximation

Before applying the normal approximation to a sample proportion, certain conditions must be met. These include independence of observations and a large enough sample size such that the expected numbers of successes (np) and failures (n(1-p)) are both at least 10. This ensures that the sampling distribution is sufficiently symmetric and spreads evenly for the approximation to be valid.

Sampling Distribution of a Proportion

This concept refers to the probability distribution of the sample proportion from repeated samples of the same size drawn from the population. When the CLT conditions are met, this distribution can be approximated by a normal distribution with a mean equal to the population proportion (p) and a standard error defined by sqrt[p(1-p)/n].

Standard Error of a Proportion

The standard error measures the variability of the sample proportion around the population proportion. It is calculated using the formula sqrt[p(1-p)/n]. This value is crucial to determine the spread of the sampling distribution and to perform subsequent probability calculations using the normal model.

Z-Score Calculation

A Z-score converts the sample proportion into the number of standard deviations it is away from the mean of the sampling distribution. This standardization allows the use of the standard normal distribution table to find probabilities related to the sample proportion, facilitating hypothesis testing and probability estimation.

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