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 40 40 39 39 38 38 41
Answers
GPT-4o mini
To determine the minimum sample size needed for using normal approximations in estimating a population proportion, we use the formula related to the sample size \( n \) for a proportion:
\[
n = \frac{(Z^2 \cdot p \cdot (1-p))}{E^2}
\]
Where:
- \( Z \) is the Z-value corresponding to the desired confidence level,
- \( p \) is the estimated proportion (probability of success),
- \( E \) is the margin of error.
However, to use the normal approximation reliably, certain conditions should be met, which typically include:
1. \( np \geq 10 \)
2. \( n(1 - p) \geq 10 \)
Given \( p = 0.25 \), we can calculate:
1. For \( np \geq 10 \):
\[
n \cdot 0.25 \geq 10 \implies n \geq \frac{10}{0.25} = 40
\]
2. For \( n(1 - p) \geq 10 \):
\[
n \cdot (1 - 0.25) \geq 10 \implies n \cdot 0.75 \geq 10 \implies n \geq \frac{10}{0.75} \approx 13.33
\]
The more stringent condition here is \( n \geq 40 \).
Thus, the minimum sample size needed to use normal approximations for an estimate about this population, given a probability of success of 25 percent, is **40**.
So, the answer is **40**.
\[
n = \frac{(Z^2 \cdot p \cdot (1-p))}{E^2}
\]
Where:
- \( Z \) is the Z-value corresponding to the desired confidence level,
- \( p \) is the estimated proportion (probability of success),
- \( E \) is the margin of error.
However, to use the normal approximation reliably, certain conditions should be met, which typically include:
1. \( np \geq 10 \)
2. \( n(1 - p) \geq 10 \)
Given \( p = 0.25 \), we can calculate:
1. For \( np \geq 10 \):
\[
n \cdot 0.25 \geq 10 \implies n \geq \frac{10}{0.25} = 40
\]
2. For \( n(1 - p) \geq 10 \):
\[
n \cdot (1 - 0.25) \geq 10 \implies n \cdot 0.75 \geq 10 \implies n \geq \frac{10}{0.75} \approx 13.33
\]
The more stringent condition here is \( n \geq 40 \).
Thus, the minimum sample size needed to use normal approximations for an estimate about this population, given a probability of success of 25 percent, is **40**.
So, the answer is **40**.