Answer:
(1) A Normal approximation to binomial can be applied for population 1, if <em>n</em> = 100.
(2) A Normal approximation to binomial can be applied for population 2, if <em>n</em> = 100, 50 and 40.
(3) A Normal approximation to binomial can be applied for population 2, if <em>n</em> = 100, 50, 40 and 20.
Step-by-step explanation:
Consider a random variable <em>X</em> following a Binomial distribution with parameters <em>n </em>and <em>p</em>.
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:
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 <em>n</em> as follows:
![n_{a}p_{1}=10\times 0.10=1](https://tex.z-dn.net/?f=n_%7Ba%7Dp_%7B1%7D%3D10%5Ctimes%200.10%3D1%3C10%5C%5C%5C%5Cn_%7Bb%7Dp_%7B1%7D%3D100%5Ctimes%200.10%3D10%3D10%5C%5C%5C%5Cn_%7Bc%7Dp_%7B1%7D%3D50%5Ctimes%200.10%3D5%3C10%5C%5C%5C%5Cn_%7Bd%7Dp_%7B1%7D%3D40%5Ctimes%200.10%3D4%3C10%5C%5C%5C%5Cn_%7Be%7Dp_%7B1%7D%3D20%5Ctimes%200.10%3D2%3C10)
Thus, a Normal approximation to binomial can be applied for population 1, if <em>n</em> = 100.
(2)
Check the Normal approximation conditions for population 2, for all the provided <em>n</em> as follows:
![n_{a}p_{1}=10\times 0.30=310\\\\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](https://tex.z-dn.net/?f=n_%7Ba%7Dp_%7B1%7D%3D10%5Ctimes%200.30%3D3%3C10%5C%5C%5C%5Cn_%7Bb%7Dp_%7B1%7D%3D100%5Ctimes%200.30%3D30%3E10%5C%5C%5C%5Cn_%7Bc%7Dp_%7B1%7D%3D50%5Ctimes%200.30%3D15%3E10%5C%5C%5C%5Cn_%7Bd%7Dp_%7B1%7D%3D40%5Ctimes%200.10%3D12%3E10%5C%5C%5C%5Cn_%7Be%7Dp_%7B1%7D%3D20%5Ctimes%200.10%3D6%3C10)
Thus, a Normal approximation to binomial can be applied for population 2, if <em>n</em> = 100, 50 and 40.
(3)
Check the Normal approximation conditions for population 3, for all the provided <em>n</em> as follows:
![n_{a}p_{1}=10\times 0.50=510\\\\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](https://tex.z-dn.net/?f=n_%7Ba%7Dp_%7B1%7D%3D10%5Ctimes%200.50%3D5%3C10%5C%5C%5C%5Cn_%7Bb%7Dp_%7B1%7D%3D100%5Ctimes%200.50%3D50%3E10%5C%5C%5C%5Cn_%7Bc%7Dp_%7B1%7D%3D50%5Ctimes%200.50%3D25%3E10%5C%5C%5C%5Cn_%7Bd%7Dp_%7B1%7D%3D40%5Ctimes%200.50%3D20%3E10%5C%5C%5C%5Cn_%7Be%7Dp_%7B1%7D%3D20%5Ctimes%200.10%3D10%3D10)
Thus, a Normal approximation to binomial can be applied for population 2, if <em>n</em> = 100, 50, 40 and 20.