To answer the question in the image:

Why is it necessary to check whether the population is approximately normal before constructing a confidence interval?

It is necessary because many statistical methods, including constructing confidence intervals for the mean, rely on the assumption that the sampling distribution of the sample mean is approximately normal. If the population distribution is normal, then the sample mean will also be normally distributed, regardless of the sample size.

For small sample sizes (typically $n < 30$), this normality assumption becomes more critical. If the population distribution is not approximately normal, the validity of the confidence interval may be compromised. However, for larger sample sizes, the Central Limit Theorem states that the sampling distribution of the sample mean will approximate normality regardless of the population's distribution.

Would you like further details on this explanation or any other assistance with the problem?

Here are 5 related questions to expand your understanding:

1. What is the Central Limit Theorem, and why is it important in statistics?
2. How does sample size influence the shape of the sampling distribution of the mean?
3. What are some methods for checking whether a population is approximately normal?
4. What happens if the population distribution is skewed or non-normal and the sample size is small?
5. Can you construct a confidence interval if the population distribution is not normal? If so, under what conditions?

Tip: When in doubt about the population's normality, consider creating a histogram or using statistical tests (e.g., Shapiro-Wilk test) to assess normality before proceeding with parametric methods.