A random sample of 40 college students has a mean earnings of​ $3120 over the summer months. Assume the population standard deviation of earnings over the summer months for college students is​ $677 and that the distribution of earnings over the summer months for college students is normally distributed. Determine whether a normal distribution or a​ t-distribution should be used or whether neither of these can be used to construct a confidence interval.
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Part 1
A.
Cannot use a normal distribution or a​ t-distribution.
B.
Use a normal distribution.
C.
Use a​ t-distribution.

To determine whether to use a normal distribution, a t-distribution, or neither to construct a confidence interval, we should consider the following factors:

1. **Sample Size**: If the sample size is large (typically $ n \geq 30 $), the Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.
2. **Population Standard Deviation ($\sigma$) Known or Unknown**: 
   - If the population standard deviation ($\sigma$) is known, and the sample size is large, a normal distribution can be used.
   - If the population standard deviation ($\sigma$) is unknown and the sample size is small, the t-distribution is typically used.

### Given Information:
- Sample size $ n = 40 $ (which is greater than 30, so the sample size is large).
- Population standard deviation $ \sigma = 677 $ is known.
- The population distribution of earnings is normally distributed.

### Conclusion:
Since the sample size is large and the population standard deviation is known, we should **use a normal distribution** to construct the confidence interval.

**Answer: B. Use a normal distribution.**

Would you like to go into more detail, or do you have any questions? Here are 8 related questions you might find interesting:

1. How does the sample size affect the choice between normal and t-distribution?
2. What is the Central Limit Theorem, and how does it apply here?
3. Can the t-distribution be used when the sample size is large and the population standard deviation is unknown?
4. How would the choice change if the population standard deviation were unknown?
5. What is the difference between a normal distribution and a t-distribution?
6. How do you construct a confidence interval using a normal distribution?
7. Why is it important that the population distribution is normal?
8. What would happen if the population distribution were not normal?

**Tip:** Always check whether the population standard deviation is known and the sample size when deciding between a normal or t-distribution.

Understand when to use a normal distribution or t-distribution for constructing a confidence interval. Example problem: determining distribution choice based on sample size, known population standard deviation, and normal distribution assumption.

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