Respuesta :
Answer:
(1) A Normal approximation to binomial can be applied for population 1, if n = 100.
(2) A Normal approximation to binomial can be applied for population 2, if n = 100, 50 and 40.
(3) A Normal approximation to binomial can be applied for population 2, if n = 100, 50, 40 and 20.
Step-by-step explanation:
Consider a random variable X following a Binomial distribution with parameters n and p.
If the sample selected is too large and the probability of success is close to 0.50 a Normal approximation to binomial can be applied to approximate the distribution of X if the following conditions are satisfied:
- np ≥ 10
- n(1 - p) ≥ 10
The three populations has the following proportions:
p₁ = 0.10
p₂ = 0.30
p₃ = 0.50
(1)
Check the Normal approximation conditions for population 1, for all the provided n as follows:
[tex]n_{a}p_{1}=10\times 0.10=1<10\\\\n_{b}p_{1}=100\times 0.10=10=10\\\\n_{c}p_{1}=50\times 0.10=5<10\\\\n_{d}p_{1}=40\times 0.10=4<10\\\\n_{e}p_{1}=20\times 0.10=2<10[/tex]
Thus, a Normal approximation to binomial can be applied for population 1, if n = 100.
(2)
Check the Normal approximation conditions for population 2, for all the provided n as follows:
[tex]n_{a}p_{1}=10\times 0.30=3<10\\\\n_{b}p_{1}=100\times 0.30=30>10\\\\n_{c}p_{1}=50\times 0.30=15>10\\\\n_{d}p_{1}=40\times 0.10=12>10\\\\n_{e}p_{1}=20\times 0.10=6<10[/tex]
Thus, a Normal approximation to binomial can be applied for population 2, if n = 100, 50 and 40.
(3)
Check the Normal approximation conditions for population 3, for all the provided n as follows:
[tex]n_{a}p_{1}=10\times 0.50=5<10\\\\n_{b}p_{1}=100\times 0.50=50>10\\\\n_{c}p_{1}=50\times 0.50=25>10\\\\n_{d}p_{1}=40\times 0.50=20>10\\\\n_{e}p_{1}=20\times 0.10=10=10[/tex]
Thus, a Normal approximation to binomial can be applied for population 2, if n = 100, 50, 40 and 20.
