Final answer:
A confidence interval (CI) is the range that is likely to contain the true value for an entire population, with its size dependent on the confidence level and sample characteristics. A 95 percent CI offers a 95 percent chance that the interval includes the true population mean. Wider CIs correspond to higher confidence levels due to increased certainty.
Step-by-step explanation:
The range on either side of an estimate that is likely to contain the true value for the whole population is known as a confidence interval (CI). A confidence interval is an interval estimate for an unknown population parameter and is determined based on several factors, including the desired confidence level, the known information about the distribution such as the standard deviation, and the characteristics of the sample like its size.
For example, if the confidence interval is (7-2.5 to 7+2.5), the calculation yields the interval of (4.5 to 9.5). If we have a 95 percent confidence level, it means, "We estimate with 95 percent confidence that the true value of the population mean is between 4.5 and 9.5." The width of the confidence interval can be adjusted by varying the confidence level or the sample size. A 90 percent confidence interval might be (67.18 to 68.82) while a 95 percent interval might be slightly wider at (67.02 to 68.98) to increase the probability that the interval contains the true population mean.
When constructing a confidence interval, such as a 90 percent confidence interval, we mean that if we were to take repeated samples and construct confidence intervals from these samples, about 90 percent of them would contain the true value of the population mean. The confidence interval is wider for higher percentages, with a 99 percent interval being wider than a 95 percent interval.
Finally, it's important to note that confidence intervals serve as a range wherein we estimate, but do not guarantee, the true population parameter like the mean to reside. This approach is foundational to inferential statistics, enabling us to make informed guesses about the population based on samples.