Multiple Choice
Which of the following best describes the relationship between a sample and a population in the context of the sampling distribution of the sample proportion?
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8. Sampling Distributions & Confidence Intervals: Proportion
Sampling Distribution of Sample Proportion
Multiple Choice
In which of the following situations can the sampling distribution of the sample proportion be safely approximated by a normal distribution?
A
When both and are at least 5
B
When both and are at least 10, that is, and
C
When both and are at least 10
D
When the sample size is greater than 30, regardless of
Verified step by step guidance1
Understand that the sampling distribution of the sample proportion \( \hat{p} \) can be approximated by a normal distribution when certain conditions related to the sample size \( n \) and the population proportion \( p \) are met.
Recall the two key conditions for the normal approximation to be valid: both \( n p \) and \( n (1 - p) \) must be sufficiently large to ensure the distribution is not too skewed.
Identify the commonly accepted rule of thumb, which states that both \( n p \geq 10 \) and \( n (1 - p) \geq 10 \) are required for the normal approximation to hold well.
Note that this condition ensures there are enough expected successes (\( n p \)) and failures (\( n (1 - p) \)) in the sample to approximate the binomial distribution with a normal distribution.
Recognize that simply having \( n > 30 \) is not sufficient if \( p \) is very close to 0 or 1, because the distribution could still be skewed; the conditions on \( n p \) and \( n (1 - p) \) are more precise criteria.
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