For each of the following sampling scenarios, determine which distribution should be used to calculate the critical value for the 90% confidence interval for the population mean.

(In the table, Z refers to a standard normal distribution, and refers to a t distribution.)

Sampling scenario

Could use either Z or t

Unclear

The sample has size 11, and it is from a population with a distribution about which we know very little.

The sample has size 19, and it is from a normally distributed population with an unknown standard deviation.

The sample has size 95, and it is from a non-normally distributed population with a known standard deviation of 75.

Let's analyze each of the provided sampling scenarios to determine whether the **Z** distribution (standard normal distribution) or the **t** distribution should be used to calculate the critical value for the 90% confidence interval for the population mean.

### Sampling Scenarios:

#### 1. **Sample size 11; Population distribution is unknown or not well-known:**
   - **Explanation:** When the sample size is small (\( n < 30 \)) and the population distribution is not well-known, the **t distribution** is the appropriate choice. The t distribution accounts for the additional uncertainty due to the small sample size and unknown population standard deviation.
   - **Answer:** **t distribution**

#### 2. **Sample size 19; Population is normally distributed with an unknown standard deviation:**
   - **Explanation:** Although the population is normally distributed, the sample size is still small (\( n < 30 \)) and the population standard deviation is unknown. Therefore, the **t distribution** is more appropriate in this case as well.
   - **Answer:** **t distribution**

#### 3. **Sample size 95; Population is non-normally distributed with a known standard deviation of 75:**
   - **Explanation:** In this case, the sample size is large (\( n \geq 30 \)), which allows the use of the Central Limit Theorem. This theorem lets us assume that the sampling distribution of the sample mean is approximately normal, regardless of the population distribution. Since the population standard deviation is known, the **Z distribution** should be used.
   - **Answer:** **Z distribution**

### Summary:
- **Scenario 1:** **t distribution**
- **Scenario 2:** **t distribution**
- **Scenario 3:** **Z distribution**

Would you like more details or have any questions about these calculations? Here are some related questions:

1. How would the answer change if the sample size in Scenario 1 was 31 instead of 11?
2. Why is the Central Limit Theorem important in determining the appropriate distribution?
3. Can you explain the difference between the Z and t distributions in more detail?
4. How would you calculate the confidence interval using the Z distribution in Scenario 3?
5. What happens if the standard deviation is unknown in a large sample?

**Tip:** For small sample sizes and unknown population standard deviations, the t distribution is generally more appropriate. As the sample size increases, the t distribution approaches the Z distribution.

Explore how to determine critical values for 90% confidence intervals in various sampling scenarios. Understand when to use the Z distribution (standard normal distribution) and when the t distribution is more appropriate. Includes detailed explanations and examples.

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