Final answer:
The question pertains to confidence intervals in statistics, a branch of Mathematics, where repeated samples provide estimates for the true population mean. The Central Limit Theorem is key in such calculations for a range of sample sizes and confidence levels.
Step-by-step explanation:
The question falls under the subject of Mathematics and concerns statistics, specifically related to confidence intervals and sampling distributions. Confidence intervals provide a range within which the true population mean is estimated to lie with a given level of confidence. The Central Limit Theorem underpins such calculations, indicating that the distribution of sample means will be approximately normal, even if the population distribution is not, provided the sample size is large enough.
In part b. of the original question, the statement should accurately be that approximately 90 percent of the confidence intervals calculated from repeated samples would contain the true value of the population mean, not the sample mean.
For question 7.1 about an unknown distribution, we would use the Central Limit Theorem and standard deviations of the sample mean distribution to find the probability of the sample mean lying within a specific range.
Regarding Example 8.7, determining the necessary sample size for a given confidence interval and margin of error involves using the formula for the standard error of the mean, which includes the population standard deviation and desired confidence level.