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To address Part 1 (a), let's discuss why checking for approximate normality is important when constructing a confidence interval.

When constructing a confidence interval, especially for means, we often rely on the assumption that the sampling distribution of the mean is approximately normal. This is crucial because many statistical procedures (like calculating precise confidence intervals) depend on the normality assumption. Specifically:

1. **For Small Sample Sizes (like 9 in this case):** If the sample size is small, we need to assume or verify that the population is approximately normal to justify using the normal distribution for inference. Without normality, the confidence interval might not be accurate.
  
2. **Central Limit Theorem (CLT):** For larger samples (usually \( n > 30 \)), the Central Limit Theorem assures us that the sampling distribution of the mean will be approximately normal regardless of the population's distribution. However, with a small sample like 9, we can't rely on the CLT alone and thus need to check for normality.

In short, it's necessary to check whether the population is approximately normal because **small samples require the population to be normal for accurate confidence interval construction**.

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

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Here are some additional related questions:
1. What are some common methods to check for normality in a dataset?
2. How would the approach differ if the sample size were larger (say, 50)?
3. How does the standard deviation of the population affect confidence interval calculations?
4. Why is the Central Limit Theorem important in inferential statistics?
5. What alternatives exist if the population is not normal and the sample size is small?

**Tip:** When working with small samples, visual checks (like histograms) or statistical tests (e.g., Shapiro-Wilk test) can help assess normality.

Explain why it is necessary to check whether the population is approximately normal before constructing a confidence interval.

Learn why it's essential to check if the population is approximately normal before constructing a confidence interval, especially for small sample sizes. This problem covers statistical concepts including the Central Limit Theorem, standard error, and the significance of normal distribution assumptions. Suitable for high school students in Grades 10-12.

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