Reincarnation Suppose 21% of all American teens (age 13–17 years) believe in reincarnation.


b. Explain why Bob’s sample of 100 randomly selected American teens might result in 18 who believe in reincarnation, while Alicia’s independent sample of 100 randomly selected American teens might result in 22 who believe in reincarnation.

Verified step by step guidance

1. Recognize that the problem involves a population proportion, where 21% of American teens believe in reincarnation, and samples of size 100 are taken independently.
2. Understand that the number of teens who believe in reincarnation in each sample can be modeled by a binomial distribution with parameters \(n = 100\) and \(p = 0.21\).
3. Recall that due to random sampling variability, the actual number of teens in each sample who believe in reincarnation can differ from the expected value \(np = 100 \times 0.21 = 21\).
4. Explain that Bob's sample resulting in 18 believers and Alicia's sample resulting in 22 believers are both plausible outcomes because the binomial distribution allows for variation around the mean, reflecting natural sampling variability.
5. Conclude that differences in sample results occur because each sample is independent and subject to chance, so the observed counts can fluctuate even if the underlying population proportion remains constant.

Key Concepts

Sampling Variability

Sampling variability refers to the natural differences that occur between samples drawn from the same population. Even if two samples are the same size and randomly selected, the number of individuals with a certain characteristic can vary due to chance. This explains why Bob’s sample might have 18 believers and Alicia’s 22, despite both sampling from the same population.

Random Sampling

Random sampling is a method where each member of the population has an equal chance of being selected. This process helps ensure that the sample represents the population fairly, but it also introduces randomness, which can cause sample results to differ from the true population proportion.

Population Proportion

The population proportion is the true fraction of individuals in the entire population with a specific characteristic, here 21% believing in reincarnation. Sample proportions estimate this value, but due to sampling variability, individual samples may have proportions slightly above or below the population proportion.