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From the image, the task seems to involve choosing the appropriate test statistic for hypothesis testing based on different sampling scenarios. Let’s break down the questions:

### Given information:
- **Population mean**: 470
- **Sample mean**: 488
- **Sample standard deviation**: 80
- **Test statistic**: We need to determine the appropriate test (z or t) for the given scenarios.

### Question (a):
- **Sample size**: 14
- **Population distribution**: Normally distributed
- **Standard deviation**: Unknown

When the sample size is small (n < 30) and the population standard deviation is unknown, **use the t-statistic**. This is because the t-distribution accounts for the uncertainty in estimating the population standard deviation when the sample size is small.

So, for **(a)**, the appropriate test is the **t-statistic**.

### Question (b):
- **Sample size**: 18
- **Population distribution**: Unknown, and we know very little about it.

In this case, the population distribution is unknown, and the sample size is again small (n = 18). Since we also do not know the population standard deviation, the **t-statistic** is again appropriate here because we need to rely on the sample data to estimate the variability.

So, for **(b)**, the appropriate test is also the **t-statistic**.

### Summary of Answers:
- **(a)**: t-statistic
- **(b)**: t-statistic

Would you like more details on how to calculate the t-statistic or any further explanation?

Here are 5 additional questions to expand on this concept:
1. What is the difference between a z-statistic and a t-statistic?
2. How does sample size affect the choice between z and t tests?
3. How does the Central Limit Theorem affect hypothesis testing?
4. When can we use the z-statistic even with an unknown population standard deviation?
5. How does the shape of the t-distribution compare to the normal distribution for small sample sizes?

**Tip**: When the sample size is greater than 30, the t-distribution approximates the z-distribution, making it easier to use the z-statistic.

We want to conduct a hypothesis test of the claim that the population mean score on a nationwide examination in anthropology is different from 470. For each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean.

Learn how to choose the correct test statistic (t-statistic or z-statistic) for hypothesis testing of the population mean score based on different sample sizes and distributions. This guide covers core statistical concepts such as hypothesis testing, the t-distribution, and normal distribution.

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