# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 16

A confidence interval is a random interval. Or a confidence interval is a function and its gives different output based on the value taken by the random variable $\, \bar X \,$. When we say 95% confidence interval - it means that 95% of the intervals generated in this way will contain the population mean $\, \mu \,$.

(a) True. The confidence interval is formed using random variable $\, \bar X \,$ as it’s center, thus the center is a random variable.

(b) True. This is as per the definition.

(c) False.

(d) False. To form 100 confidence intervals, it means either we will need a different random variable $\, \bar X \,$ or we will need different population parameter instead of $\, \mu \,$. But the confidence interval talks about a particular parameter and a particular random variable used for estimating that parameter. Also, note that it might be the case that when in the problem, a confidence interval is mentioned - it means a particular interval instead of random interval. In this case also the answer is false because it might be the case that all the 100 intervals formed might belong to the 5% set of the intervals that do not contain population mean. On average the answer is true but it is not true for every set of 100 intervals.

$$\tag*{\blacksquare}$$