Question

1. a school has 500 students: 300 juniors and 200 seniors. a sample of 50 is needed. how many juniors and seniors should be selected?
2. a company has 200 staff: 100 male and 100 female. a sample of 40 is needed. how many from each group?
3. a population has 1000 individuals divided into three groups: a = 500, b = 300, c = 200. a sample of 100 is needed. allocate the sample.

Explanation:

Step1: Calculate the proportion for each group

For problem 4:
The proportion of juniors is $\frac{300}{500}=\frac{3}{5}$, and the proportion of seniors is $\frac{200}{500}=\frac{2}{5}$.
For problem 5:
The proportion of males is $\frac{100}{200}=\frac{1}{2}$, and the proportion of females is $\frac{100}{200}=\frac{1}{2}$.
For problem 6:
The proportion of group A is $\frac{500}{1000}=\frac{1}{2}$, the proportion of group B is $\frac{300}{1000}=\frac{3}{10}$, and the proportion of group C is $\frac{200}{1000}=\frac{1}{5}$.

Step2: Multiply the proportion by the sample size

For problem 4:
Number of juniors in sample: $50\times\frac{3}{5} = 30$
Number of seniors in sample: $50\times\frac{2}{5}=20$
For problem 5:
Number of males in sample: $40\times\frac{1}{2}=20$
Number of females in sample: $40\times\frac{1}{2}=20$
For problem 6:
Number of individuals from group A in sample: $100\times\frac{1}{2}=50$
Number of individuals from group B in sample: $100\times\frac{3}{10}=30$
Number of individuals from group C in sample: $100\times\frac{1}{5}=20$

Answer:

1. 30 juniors and 20 seniors.
2. 20 males and 20 females.
3. 50 from group A, 30 from group B, and 20 from group C.