Notes on Simple Random Sampling

Chapter 1.3: Simple Random Sampling

Objectives

• Understand and obtain a simple random sample.

Definition of Simple Random Sampling

• A sample of size $n$ from a population of size $N$ is achieved through simple random sampling if all possible samples of size $n$ have an equally likely chance of being selected.

  • The resulting sample is referred to as a simple random sample.

Example of Simple Random Sampling

Scenario

• Context: Sophia has 4 tickets to a concert.

• Population: 6 friends who want to attend the concert are:

  • Yolanda

  • Michael

  • Kevin

  • Marissa

  • Annie

  • Katie

Procedure

1. List of Possible Samples: Determine all possible combinations of size $n = 3$ from the population of size $N = 6$.

   • Each selection is a group of 3 friends chosen from the 6 without replacement.

   • Example of one possible sample: Yolanda, Michael, Kevin.

2. Likelihood of a Specific Sample: Assess the probability of selecting the particular sample of Michael, Kevin, and Marissa.

   • Note: The problem explicitly states to evaluate the likelihood of this specific combination occurring within the context of simple random sampling. The calculations for this likelihood would be based on the total combinations possible.

Calculation of Possible Combinations

• The number of ways to choose a sample of size $n$ from a population of size $N$ can be calculated using the binomial coefficient, represented as:
  $\binom{N}{n} = \frac{N!}{n!(N-n)!}$

• For this example where $N = 6$ and $n = 3$:
  $\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$

Conclusion

• There are 20 possible samples of size 3 from 6 friends, and each of these samples has an equal chance of being selected.