(b) For each of the following sampling scenarios, determine which distribution should be used to calculate the critical value for the 99% confidence interval for the population mean. (In the table, Z refers to a standard normal distribution, and t refers to a *t* distribution.) Sampling scenario  options: Z or t or "Could use Z either T" or Unclear.

(1)The sample has size 80, and it is from a non-normally distributed population with a known standard deviation of 7719.

(2)The sample has size 10, and it is from a normally distributed population with an unknown standard deviation.

(3)The sample has size 95, and it is from a non-normally distributed population.

To determine which distribution to use for the critical value in each scenario, consider the following factors:

1. **Sample size (n):** If the sample size is large (typically \( n \geq 30 \)), the Central Limit Theorem allows us to use the \( Z \)-distribution even if the population is not normally distributed, provided the population standard deviation is known. For smaller sample sizes, the \( t \)-distribution is usually used, especially when the population standard deviation is unknown.
   
2. **Population distribution:** If the population is normally distributed, the \( t \)-distribution is used when the sample size is small and the population standard deviation is unknown. If the population is not normally distributed and the sample size is small, the use of the \( t \)-distribution is more complex.

3. **Known or unknown population standard deviation:** If the population standard deviation is known, the \( Z \)-distribution is used, regardless of the sample size.

Now, let's apply this to each scenario:

1. **Scenario 1: The sample has size 80, and it is from a non-normally distributed population with a known standard deviation of 7719.**
   - **Distribution to use:** \( Z \)-distribution.
   - **Reason:** The sample size is large (\( n = 80 \)), and the population standard deviation is known. Even though the population is non-normally distributed, the large sample size allows us to use the \( Z \)-distribution.

2. **Scenario 2: The sample has size 10, and it is from a normally distributed population with an unknown standard deviation.**
   - **Distribution to use:** \( t \)-distribution.
   - **Reason:** The sample size is small (\( n = 10 \)), the population is normally distributed, and the population standard deviation is unknown. The \( t \)-distribution is appropriate in this case.

3. **Scenario 3: The sample has size 95, and it is from a non-normally distributed population.**
   - **Distribution to use:** \( Z \)-distribution.
   - **Reason:** The sample size is large (\( n = 95 \)), which allows the use of the \( Z \)-distribution even though the population is non-normally distributed.

### Summary Table:
| Sampling Scenario                                               | Distribution to Use |
|-----------------------------------------------------------------|---------------------|
| (1) \( n = 80 \), non-normal population, known \( \sigma = 7719 \) | \( Z \)             |
| (2) \( n = 10 \), normal population, unknown \( \sigma \)       | \( t \)             |
| (3) \( n = 95 \), non-normal population                         | \( Z \)             |

Would you like more details on these scenarios, or do you have any questions?

### Related Questions:
1. What is the difference between the \( Z \)-distribution and the \( t \)-distribution?
2. How does the shape of the \( t \)-distribution change with the sample size?
3. When should you use a \( t \)-distribution instead of a \( Z \)-distribution?
4. What assumptions are necessary for using the \( t \)-distribution?
5. How do you calculate the critical value from a \( Z \)-distribution?
6. How does knowing the population standard deviation affect confidence intervals?
7. What would happen if you used the wrong distribution for your confidence interval?
8. How can you verify whether a population is normally distributed?

**Tip:** When the sample size is large, the \( t \)-distribution approaches the \( Z \)-distribution, so the difference between using \( t \) and \( Z \) becomes negligible.

Learn how to determine the appropriate distribution (Z or t) for calculating critical values in confidence intervals based on sample size and population characteristics. Explore scenarios including non-normal populations with known and unknown standard deviations.

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