Question
What is the minimum sample size needed in order to use normal approximations for an estimate about a population, given a probability of success of 25 percent?(1 point)
Responses
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39
Responses
38
38
41
41
40
40
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39
Answers
GPT-4o mini
To determine the minimum sample size needed for using normal approximations for a proportion, you can use the rule of thumb involving both the probability of success (p) and the probability of failure (q), where \( q = 1 - p \).
The conditions for using the normal approximation are typically that both \( np \) and \( nq \) should be greater than or equal to 5.
Given that the probability of success \( p = 0.25 \), we can calculate \( q = 1 - p = 0.75 \).
We want both:
1. \( np \geq 5 \)
2. \( nq \geq 5 \)
This gives us the following inequalities:
1. \( n(0.25) \geq 5 \) which simplifies to \( n \geq \frac{5}{0.25} = 20 \)
2. \( n(0.75) \geq 5 \) which simplifies to \( n \geq \frac{5}{0.75} \approx 6.67 \)
The more stringent requirement is \( n \geq 20 \).
However, to ensure that both \( np \) and \( nq \) are comfortably above 5, it's common to choose a slightly larger sample size. A typical recommendation is to use a sample size of at least 30 or even 40. Given the options provided (38, 41, 40, 39), a safe choice that satisfies the conditions is 40.
Therefore, the minimum sample size needed to use normal approximations for an estimate about a population, given a probability of success of 25 percent, is **40**.
The conditions for using the normal approximation are typically that both \( np \) and \( nq \) should be greater than or equal to 5.
Given that the probability of success \( p = 0.25 \), we can calculate \( q = 1 - p = 0.75 \).
We want both:
1. \( np \geq 5 \)
2. \( nq \geq 5 \)
This gives us the following inequalities:
1. \( n(0.25) \geq 5 \) which simplifies to \( n \geq \frac{5}{0.25} = 20 \)
2. \( n(0.75) \geq 5 \) which simplifies to \( n \geq \frac{5}{0.75} \approx 6.67 \)
The more stringent requirement is \( n \geq 20 \).
However, to ensure that both \( np \) and \( nq \) are comfortably above 5, it's common to choose a slightly larger sample size. A typical recommendation is to use a sample size of at least 30 or even 40. Given the options provided (38, 41, 40, 39), a safe choice that satisfies the conditions is 40.
Therefore, the minimum sample size needed to use normal approximations for an estimate about a population, given a probability of success of 25 percent, is **40**.