Guided Practice: Simple Random Sampling and Sample Enumeration

Q1. List all possible samples of size n = 3 from a population of size N = 6 (Yolanda, Michael, Kevin, Marissa, Annie, Katie). Once an individual is chosen, they cannot be chosen again.

Background

Topic: Simple Random Sampling

This question tests your understanding of how to enumerate all possible simple random samples of a given size from a finite population, and the concept that each sample has an equal chance of being selected.

Key Terms and Formulas

• Simple Random Sample: A sample in which every possible group of n individuals from the population has an equal chance of being selected.

• Combination Formula: The number of ways to choose n items from N without regard to order is given by:

$C(N, n) = \frac{N!}{n!(N-n)!}$

• Where $N!$ means "N factorial," the product of all positive integers up to N.

Step-by-Step Guidance

1. Identify the population: The six friends are Yolanda, Michael, Kevin, Marissa, Annie, and Katie.

2. Determine the sample size: You are to select n = 3 friends from N = 6.

3. Use the combination formula to find out how many unique samples of 3 can be formed from 6 individuals:

   $C(6, 3) = \frac{6!}{3!\,3!}$

4. List all possible combinations (samples) of 3 friends. Remember, the order does not matter, and no individual can be chosen more than once per sample.

5. Write out each unique group of 3 friends. For example, one sample is Yolanda, Michael, Kevin. Continue this process until all possible samples are listed.

Try solving on your own before revealing the answer!

Q2. Comment on the likelihood of the sample containing Michael, Kevin, and Marissa.

Background

Topic: Probability in Simple Random Sampling

This question asks you to consider the probability that a specific group (Michael, Kevin, and Marissa) is selected as the sample, given that all samples are equally likely.

Key Terms and Formulas

• Probability of a Specific Sample: In simple random sampling, the probability of selecting any particular sample is:

$P(\text{specific sample}) = \frac{1}{\text{total number of samples}}$

Step-by-Step Guidance

1. Recall from the previous part the total number of possible samples of size 3 from 6 individuals.

2. Recognize that each possible sample (including the one with Michael, Kevin, and Marissa) has an equal chance of being selected.

3. Set up the probability formula for selecting the specific sample (Michael, Kevin, and Marissa):

   $P = \frac{1}{C(6, 3)}$

4. Substitute the value of $C(6, 3)$ from earlier to express the probability numerically, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answers

Q1: There are 20 possible samples of size 3 from 6 individuals. The full list includes all unique combinations, such as (Yolanda, Michael, Kevin), (Yolanda, Michael, Marissa), etc.

Q2: The probability of selecting the specific sample (Michael, Kevin, Marissa) is $\frac{1}{20}$, since there are 20 equally likely samples.

This demonstrates that in simple random sampling, every possible group of the specified size has the same chance of being chosen.