Describe the circumstances under which the shape of the sampling distribution of p̂ is approximately normal.

Verified step by step guidance

1. Understand that \( \hat{p} \) represents the sample proportion, which is a random variable based on the outcomes of a sample from a population.
2. Recall that the sampling distribution of \( \hat{p} \) describes the distribution of sample proportions over many repeated samples of the same size from the population.
3. The shape of the sampling distribution of \( \hat{p} \) is approximately normal when the sample size \( n \) is sufficiently large, and the population proportion \( p \) is not too close to 0 or 1.
4. Specifically, the normal approximation is appropriate if both \( n \times p \geq 10 \) and \( n \times (1 - p) \geq 10 \). These conditions ensure that there are enough expected successes and failures in the sample for the distribution to be symmetric and bell-shaped.
5. When these conditions are met, the Central Limit Theorem applies, and the sampling distribution of \( \hat{p} \) can be approximated by a normal distribution with mean \( p \) and standard deviation \( \sqrt{\frac{p(1-p)}{n}} \).

Key Concepts

Sampling Distribution of p̂

The sampling distribution of p̂ refers to the probability distribution of the sample proportion calculated from repeated random samples of the same size from a population. It shows how p̂ varies from sample to sample and is fundamental for making inferences about the population proportion.

Normal Approximation to the Sampling Distribution

The sampling distribution of p̂ is approximately normal when the sample size is large enough, allowing the Central Limit Theorem to apply. This means the distribution of p̂ will be symmetric and bell-shaped, enabling the use of normal probability methods for inference.

Conditions for Normality: np and n(1-p)

For the sampling distribution of p̂ to be approximately normal, both np and n(1-p) must be at least 10, where n is the sample size and p is the population proportion. This ensures enough expected successes and failures to justify the normal approximation.