To determine which distribution to use for calculating the critical value for the 99% confidence interval in each scenario, we need to consider the sample size, the distribution of the population, and whether the population standard deviation is known or unknown. Here’s how to approach each case:

(a) The sample has a size of 105, and it is from a non-normally distributed population.

• Distribution to use: Z-distribution
  • Reasoning: Since the sample size is large ($n > 30$), the Central Limit Theorem applies, which allows us to use the Z-distribution even though the population is non-normally distributed.

(b) The sample has a size of 80, and it is from a non-normally distributed population with a known standard deviation of 77.

• Distribution to use: Z-distribution
  • Reasoning: Again, the sample size is large ($n > 30$), and the population standard deviation is known. This justifies using the Z-distribution.

(c) The sample has a size of 15, and it is from a normally distributed population with an unknown standard deviation.

• Distribution to use: t-distribution
  • Reasoning: The sample size is small ($n < 30$), and the population standard deviation is unknown. Since the population is normally distributed, the t-distribution should be used.

Would you like to discuss any of these scenarios in more detail or explore how to calculate the confidence intervals using these distributions?

Here are some related questions:

1. What are the key differences between the Z-distribution and the t-distribution?
2. Why does the Central Limit Theorem allow the use of the Z-distribution for large samples?
3. How does the shape of the t-distribution change as sample size increases?
4. What is the formula for calculating the confidence interval using the t-distribution?
5. In what situations is it appropriate to use a Z-distribution for small samples?
6. How does the known or unknown population standard deviation affect the choice of distribution?
7. Why is it important to check whether the population is normally distributed in small samples?
8. How do degrees of freedom impact the t-distribution?

Tip: When dealing with small samples, always check the normality of the population before choosing the t-distribution.