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Chapter 7: Problem 53

According to the central limit theorem, the sampling distribution of \(\hat{p}\) is approximately normal when the sample is large. What is considered a large sample in the case of the proportion? Briefly explain.

Short Answer

Expert verified
When dealing with sample proportions, a sample size is generally considered 'large' enough for the Central Limit Theorem to apply if both \(np\) and \(n(1-p)\) are greater than or equal to 10.

Step by step solution

01

Identify the concepts

In the context of sampling, a proportion (\(\hat{p}\)) is defined as the ratio of the number of sampled individuals possessing a specified characteristic to the total size of the sample. The Central Limit Theorem (CLT) applies to proportions and states that, as the sample size increases, the distribution of sample proportions approximates a normal distribution.
02

Define a large sample in terms of proportions

While there's no universally approved threshold for a 'large' sample size that applies to every scenario, a common rule of thumb is that a sample is deemed large enough if both \(np\) and \(n(1-p)\) are greater than or equal to 10, where \(n\) represents the sample size and \(p\) is the proportion under consideration. This guideline ensures that the sampling distribution of \(\hat{p}\) will be approximately normal.
03

Synthesis

Therefore, when dealing with proportions, a 'large' sample size is one where both \(np\) and \(n(1-p)\) are greater than or equal to 10.

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