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.