Final answer:
The width of a confidence interval for a population mean is most likely to be smaller if the sample size is larger and the degree of confidence is lower, because a larger sample size reduces error and variability, and a lower confidence level requires less area under the normal distribution curve.
Step-by-step explanation:
The width of a confidence interval for a population mean is most likely to be smaller if the sample size is larger and the degree of confidence is lower. This is because increasing the sample size decreases the error bound, hence narrowing the confidence interval. Conversely, lowering the degree of confidence decreases the amount of area under the normal distribution curve that we need to capture the true population mean, which results in a smaller interval.
For example, a 90% confidence interval is narrower than a 95% confidence interval because it excludes a larger percentage of the distribution. The relationship between sample size, degree of confidence, and the width of confidence intervals is a fundamental concept in statistics. An increase in sample size generally leads to increased accuracy in estimating the population mean, thus requiring a narrower interval for the same degree of confidence.
When considering whether a sample mean will fall within a certain confidence interval, it's essential to understand that with a larger sample size and decreased confidence level, the interval needed to potentially include the true population mean will be smaller due to reduced variability and required area under the curve respectively.