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%
A. 39
B. 38
C. 41
D. 40
A. 39
B. 38
C. 41
D. 40
Answers
GPT-4o mini
To use the normal approximation for a binomial distribution, we typically need both \(np\) and \(n(1-p)\) to be at least 5, where \(n\) is the sample size and \(p\) is the probability of success.
In this case, the probability of success \(p\) is 0.25 (or 25%) and the probability of failure \(q\) is \(1 - p = 0.75\).
We can set up the inequalities:
1. \(np \geq 5\)
2. \(n(1 - p) \geq 5\)
Substituting \(p\) and \(q\) into these equations:
1. \(n(0.25) \geq 5\)
2. \(n(0.75) \geq 5\)
Solving the first inequality:
\[
0.25n \geq 5 \quad \Rightarrow \quad n \geq \frac{5}{0.25} = 20
\]
Now solving the second inequality:
\[
0.75n \geq 5 \quad \Rightarrow \quad n \geq \frac{5}{0.75} \approx 6.67
\]
Since we need \(n\) to satisfy both inequalities, we take the larger value, which is \(n \geq 20\).
However, the question presents potential options, so we should check if the choices listed could provide a minimum sample size that satisfies the normal approximation criteria.
From the inequalities:
- We found \(n \geq 20\) is satisfactory based on the condition needed to apply the normal approximation.
The most probable consideration for those options (given their values around or above 20) could be understood in a different context or based on additional criteria (for example, the minimum sample size that allows a more precise approximation). The correct population sample size would depend on the context.
However, if we had to choose from the provided options (39, 38, 41, 40) directly matching with basic results could imply that the final sample could lead to precise estimates.
Nonetheless, all values presented exceed the base sample size requirement. Hence, there might not be one specific answer unless context or additional constraints are applied.
Thus, assuming a choice among the options for practical purposes is necessary, I would recommend **the option with the next reasonable confidence threshold above traditional estimates**.
Therefore, the answer is somewhat subjective based on chosen criteria, but from a strict understanding of calculation, all listed options satisfy the condition but may have different proportions for small sample adjustments.
Therefore, any higher rounds closer to values specified may suit the broader approximation context.
Given the four choices presented, as traditionally, higher is generally chosen for conservative estimates, perhaps answer **D. 40** or **C. 41** would suit under broader conditions for estimates ensuring certainty, bearing in mind statistical practices.
In this case, the probability of success \(p\) is 0.25 (or 25%) and the probability of failure \(q\) is \(1 - p = 0.75\).
We can set up the inequalities:
1. \(np \geq 5\)
2. \(n(1 - p) \geq 5\)
Substituting \(p\) and \(q\) into these equations:
1. \(n(0.25) \geq 5\)
2. \(n(0.75) \geq 5\)
Solving the first inequality:
\[
0.25n \geq 5 \quad \Rightarrow \quad n \geq \frac{5}{0.25} = 20
\]
Now solving the second inequality:
\[
0.75n \geq 5 \quad \Rightarrow \quad n \geq \frac{5}{0.75} \approx 6.67
\]
Since we need \(n\) to satisfy both inequalities, we take the larger value, which is \(n \geq 20\).
However, the question presents potential options, so we should check if the choices listed could provide a minimum sample size that satisfies the normal approximation criteria.
From the inequalities:
- We found \(n \geq 20\) is satisfactory based on the condition needed to apply the normal approximation.
The most probable consideration for those options (given their values around or above 20) could be understood in a different context or based on additional criteria (for example, the minimum sample size that allows a more precise approximation). The correct population sample size would depend on the context.
However, if we had to choose from the provided options (39, 38, 41, 40) directly matching with basic results could imply that the final sample could lead to precise estimates.
Nonetheless, all values presented exceed the base sample size requirement. Hence, there might not be one specific answer unless context or additional constraints are applied.
Thus, assuming a choice among the options for practical purposes is necessary, I would recommend **the option with the next reasonable confidence threshold above traditional estimates**.
Therefore, the answer is somewhat subjective based on chosen criteria, but from a strict understanding of calculation, all listed options satisfy the condition but may have different proportions for small sample adjustments.
Therefore, any higher rounds closer to values specified may suit the broader approximation context.
Given the four choices presented, as traditionally, higher is generally chosen for conservative estimates, perhaps answer **D. 40** or **C. 41** would suit under broader conditions for estimates ensuring certainty, bearing in mind statistical practices.