Peanut and tree nut allergies are considered to be the most serious food allergies. According to the National Institute of Allergy and Infectious Diseases, roughly 1% of Americans are allergic to peanuts or tree nuts. A random sample of 1500 Americans is obtained.
a. Explain why a large sample is needed for the distribution of the sample proportion to be approximately normal.

Verified step by step guidance

1. Identify the population proportion \(p\) given in the problem, which is 1%, or \(p = 0.01\).
2. Recall that the sampling distribution of the sample proportion \(\hat{p}\) is approximately normal if the sample size \(n\) is large enough to satisfy the conditions \(n p \geq 10\) and \(n (1 - p) \geq 10\).
3. Calculate \(n p\) and \(n (1 - p)\) using the sample size \(n = 1500\) and the population proportion \(p = 0.01\) to check if these conditions hold.
4. Explain that these conditions ensure there are enough expected successes and failures in the sample, which allows the Central Limit Theorem to apply and the sampling distribution of \(\hat{p}\) to be approximately normal.
5. Conclude that a large sample size is needed to meet these conditions, making the normal approximation valid for inference about the population proportion.

Key Concepts

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the distribution of proportions calculated from many random samples of the same size. It helps us understand how the sample proportion varies from sample to sample and is essential for making inferences about the population proportion.

Normal Approximation to the Binomial Distribution

When the sample size is large, the distribution of the sample proportion can be approximated by a normal distribution due to the Central Limit Theorem. This approximation is valid if both np and n(1-p) are at least 10, ensuring the sample proportion's distribution is roughly symmetric and bell-shaped.

Importance of Large Sample Size

A large sample size reduces variability and ensures the conditions for the normal approximation are met. It allows the sample proportion to be more stable and the sampling distribution to be approximately normal, which is crucial for applying statistical methods like confidence intervals and hypothesis tests.