Multiple Choice
In the context of the sampling distribution of the sample proportion , why is a sample typically used instead of collecting data from the entire population?
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8. Sampling Distributions & Confidence Intervals: Proportion
Sampling Distribution of Sample Proportion
Multiple Choice
Which of the following is not a property of the sampling distribution of the sample proportion?
A
The mean of the sampling distribution equals the population proportion .
B
The sampling distribution of the sample proportion is always perfectly normal, regardless of sample size.
C
The sample proportion is an unbiased estimator of the population proportion .
D
The standard deviation of the sampling distribution is given by .
Verified step by step guidance1
Understand that the sampling distribution of the sample proportion \( \hat{p} \) describes the distribution of sample proportions from all possible samples of size \( n \) drawn from a population with true proportion \( p \).
Recall the key properties of the sampling distribution of \( \hat{p} \):
1. The mean of the sampling distribution is equal to the population proportion \( p \), meaning \( E(\hat{p}) = p \). This shows \( \hat{p} \) is an unbiased estimator.
2. The standard deviation (also called the standard error) of the sampling distribution is given by \( \sqrt{\frac{p(1-p)}{n}} \), which depends on the population proportion \( p \) and the sample size \( n \).
3. The shape of the sampling distribution of \( \hat{p} \) approaches a normal distribution as the sample size \( n \) becomes large, according to the Central Limit Theorem. However, it is not always perfectly normal for any sample size; small samples or extreme values of \( p \) can cause the distribution to be skewed.
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