Final answer:
For a 90% confidence interval with n=9 and a skewed population, the central limit theorem might not apply well due to the sample size and skewness. If the theorem is deemed applicable, a t-distribution with 8 degrees of freedom would be used. Otherwise, non-parametric methods or bootstrapping are recommended.
Step-by-step explanation:
To calculate a 90% confidence interval for a mean μ when the population is skewed, and the sample size is small (n=9), neither the normal distribution nor the standard t-distribution may apply since the skewness can greatly affect the small sample. However, if we assume that the central limit theorem applies and the sample size n=9 is sufficient for this theorem to take effect, we would use the t-distribution since the sample size is less than 30, and the population standard deviation is unknown.
The degrees of freedom (df) for this study would be n-1, which in this case would be 9-1=8. To find the critical t-value for a 90% confidence interval, we refer to a t-table or use statistical software, checking for df=8.
However, due to the population being very skewed, it is critical to assess the robustness of the central limit theorem for this small sample size. The skewed population means that the standard error might not represent the real variability accurately, leading to a potentially misleading confidence interval. In such cases, non-parametric methods or bootstrapping may provide more reliable interval estimates.