Final answer:
The questions pertain to defining random variables, identifying their possible distributions, and constructing confidence intervals using methods from statistics, which are likely part of a high school or introductory college curriculum in mathematics covering topics like normal distribution and statistical confidence.
Step-by-step explanation:
The student's questions seem to relate to probability and statistics, specifically the normal distribution, construction of confidence intervals, and understanding random variables. These are topics typically covered in high school and introductory college statistics courses.
List of values X may take on
In the context of a probability problem, the values that X may take on depend on the type of variable X represents. If X is defined as a count of discrete events, it might only take on integer values. If X represents a measurement that can vary continuously, then it could take any real number within a specified range.
Distribution of X
To give the distribution of X (X~), we would need more specific information about the variable. For example, if X represents IQ scores, we might say X follows a normal distribution with a mean (μ) and standard deviation (σ), expressed as X~N(μ, σ).
Construction of Confidence Interval
To construct a confidence interval for a population mean, you would use standard statistical methods. Assuming the sample data is normally distributed, the steps involve calculating the sample mean, determining the standard error, and using a t-distribution or z-distribution to find the critical values for the specified confidence level. The error bound, or margin of error, would provide the range within which we can be certain the population mean lies with the given level of confidence.
- State the confidence interval: Express as (sample mean - error bound, sample mean + error bound).
- Sketch the graph: Illustrate the normal curve with the confidence interval marked on the x-axis.
- Calculate the error bound: Find the product of the critical value and the standard error.