(a)The sample has size 14, and it is from a normally distributed population with a known standard deviation of 3. (b)The sample has size 13, and it is from a normally distributed population with an unknown standard deviation. (c)The sample has size 105, and it is from a non-normally distributed population.

Let's analyze each scenario to determine which distribution (Z or *t*) should be used to calculate the critical value for a 99% confidence interval.

### (a) Sample size 14, normally distributed population, known standard deviation of 3.
- **Sample size**: \( n = 14 \) (small sample).
- **Population distribution**: Normally distributed.
- **Standard deviation**: Known (\( \sigma = 3 \)).

**Conclusion**: Since the population standard deviation is known, even with a small sample size, the **Z-distribution** should be used.

### (b) Sample size 13, normally distributed population, unknown standard deviation.
- **Sample size**: \( n = 13 \) (small sample).
- **Population distribution**: Normally distributed.
- **Standard deviation**: Unknown.

**Conclusion**: Since the population standard deviation is unknown and the sample size is small, the **t-distribution** should be used.

### (c) Sample size 105, non-normally distributed population.
- **Sample size**: \( n = 105 \) (large sample).
- **Population distribution**: Non-normally distributed.
- **Standard deviation**: Not mentioned (but irrelevant given the sample size).

**Conclusion**: With a large sample size (\( n \geq 30 \)), the Central Limit Theorem allows us to assume that the sample mean is approximately normally distributed, even if the population is not. Therefore, the **Z-distribution** should be used.

### Summary
1. **Scenario (a)**: Use **Z-distribution**.
2. **Scenario (b)**: Use **t-distribution**.
3. **Scenario (c)**: Use **Z-distribution**.

Would you like any additional details or have any questions?

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Here are some related questions for further understanding:

1. How does the Central Limit Theorem justify using the Z-distribution for large sample sizes?
2. What is the impact of using the wrong distribution when calculating confidence intervals?
3. How do degrees of freedom affect the shape of the *t*-distribution?
4. In what situations would the *t*-distribution become indistinguishable from the Z-distribution?
5. What are the assumptions behind using the Z-distribution for confidence intervals?
6. How do you interpret a 99% confidence interval?
7. What is the formula for calculating the confidence interval using the Z-distribution?
8. How does a non-normal population distribution affect the choice of distribution for small sample sizes?

**Tip**: Remember that the choice between the *t*-distribution and the Z-distribution depends not just on sample size, but also on whether the population standard deviation is known.

Learn how to choose between Z-distribution and t-distribution for confidence intervals in statistics. Understand scenarios where each distribution is appropriate based on sample size, population distribution, and standard deviation. Suitable for undergraduate-level statistics students.

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