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


f. What minimum sample size would you require in order for the distribution of the sample proportion to be modeled by the normal distribution?

Verified step by step guidance

1. Identify the conditions for the sample proportion \( \hat{p} \) to be approximately normally distributed. These conditions are that both \( np \) and \( n(1-p) \) must be at least 10, where \( n \) is the sample size and \( p \) is the population proportion.
2. Write down the inequalities based on the conditions: \( np \geq 10 \) and \( n(1-p) \geq 10 \).
3. Substitute the given population proportion \( p = 0.21 \) into the inequalities: \( n \times 0.21 \geq 10 \) and \( n \times (1 - 0.21) \geq 10 \).
4. Solve each inequality for \( n \): \( n \geq \frac{10}{0.21} \) and \( n \geq \frac{10}{0.79} \).
5. Determine the minimum sample size \( n \) by taking the larger value from the two results obtained, ensuring both conditions are satisfied for the normal approximation to be valid.

Key Concepts

Sample Proportion and Its Distribution

The sample proportion is the fraction of individuals in a sample with a particular characteristic, here belief in reincarnation. Its distribution describes how this proportion varies across different samples. Understanding this helps in approximating the behavior of the sample proportion when drawing conclusions about the population.

Normal Approximation to the Binomial Distribution

When the sample size is large enough, the distribution of the sample proportion can be approximated by a normal distribution. This approximation simplifies calculations and inference. The rule of thumb requires both np and n(1-p) to be at least 10, where n is sample size and p is population proportion.

Determining Minimum Sample Size for Normal Approximation

To use the normal model for the sample proportion, the sample size must be large enough so that np ≥ 10 and n(1-p) ≥ 10. This ensures the distribution is approximately normal. Solving these inequalities for n gives the minimum sample size needed for valid normal approximation.