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

39
39

41
41

40
40

38

Answer 1

To determine the minimum sample size needed to use normal approximations for a proportion, we often use the rule of thumb that both \( np \) and \( n(1 - p) \) should be at least 10, where \( p \) is the probability of success and \( n \) is the sample size.

Given \( p = 0.25 \):

1. Calculate \( np \):
\[
np = n \times 0.25
\]

2. Calculate \( n(1-p) \):
\[
n(1 - p) = n \times (1 - 0.25) = n \times 0.75
\]

We need both \( np \geq 10 \) and \( n(1 - p) \geq 10 \).

From \( np \geq 10 \):
\[
n \times 0.25 \geq 10 \implies n \geq \frac{10}{0.25} = 40
\]

From \( n(1 - p) \geq 10 \):
\[
n \times 0.75 \geq 10 \implies n \geq \frac{10}{0.75} \approx 13.33
\]

The more restrictive condition is \( n \geq 40 \).

Thus, the minimum sample size needed is **40**.

The answer is **40**.