Final answer:
The statement about the time needed to complete an exam being a continuous random variable is true. Random variables represent values subject to chance and can exhibit different distributions like uniform or normal. Confidence intervals are used to estimate the mean of a population with varying error bounds based on the level of confidence.
Step-by-step explanation:
The statement The length of time X, needed by students in a particular course to complete a 1-hour exam, is a random variable with PDF given by f(x) is true. A random variable in mathematics, especially in statistics, is a variable whose value is subject to variations due to chance. In this context, time X is indeed a continuous random variable because it represents the duration of time which can be measured in infinitely many ways.
For instance, consider the example where the time it takes a student to finish a quiz is uniformly distributed between six and fifteen minutes, inclusive. Here, X represents the time, in minutes, it takes a student to finish a quiz, and we denote it as X ~ U(6, 15), indicating that it follows a uniform distribution. With a uniform distribution, all outcomes in the range are equally likely, and the PDF (probability density function) is constant over the range of possible times (from 6 to 15 minutes).
The context of the question also hints that X may be subject to different distributions depending on the situation, such as a normal distribution for the duration of criminal trials with known means and standard deviations, or a uniform distribution for lengths of papers or duration of trials. To draw inferences about the population mean length of time, for example, one would typically compute a confidence interval, which provides an estimated range of values that is likely to include the true mean of the population with a specified level of confidence. The width of this interval, known as the error bound, would change with different confidence levels because higher confidence requires a wider interval to ensure that it encompasses the true population parameter.