Final answer:
The statement regarding confidence intervals is true: population parameters remain constant and the interval itself is random. Confidence intervals vary with each new sample because they are based on the sample data, which is subject to randomness. The chosen confidence level directly influences the size of the interval.
Step-by-step explanation:
In response to the statement, 'In the confidence interval, the population parameter remains constant and the interval is random,' the answer is a. True. The population parameter, such as a population mean, is a fixed value that does not change. A confidence interval represents the range in which we expect the population parameter to fall, based on sample data. As we take different samples and calculate confidence intervals from them, those intervals will vary due to sampling variability. This randomness of the interval refers to the fact that it is based on the sample data which changes every time we take a new sample.
When constructing a confidence interval (CI), the desired confidence level influences the width of the interval. For example, a 90% confidence interval means that if we were to take repeated samples and construct confidence intervals from each, approximately 90% of those intervals would contain the true population parameter.
If we increase the confidence level, this would widen the interval due to needing more area under the normal distribution curve to capture that higher level of confidence. Conversely, if we opt for a lower confidence level, the interval would get narrower. Similarly, smaller sample sizes contribute to more variability, and hence, a larger interval might be needed to maintain the same level of confidence when estimating the true population mean.