As a statistical expert with a deep understanding of confidence intervals, I can provide a comprehensive explanation on the topic. Confidence intervals are a fundamental concept in statistics, used to estimate the range within which an unknown population parameter is likely to fall. They are expressed as a range and are often used to indicate the precision of an estimate.
When statisticians talk about a "90% confidence interval" or a "95% confidence interval," they are referring to the level of confidence that the true population parameter lies within the calculated range. The percentage indicates the level of confidence that the interval will contain the true parameter if the experiment is repeated an infinite number of times. It's important to note that the confidence level is not a measure of the probability that the parameter is in the interval; rather, it's the proportion of all possible intervals constructed from all possible samples that would contain the true value.
Now, regarding the question of whether a 90% confidence interval is narrower than a 95% confidence interval, the answer is generally yes, but with some nuances. The level of confidence is inversely related to the width of the confidence interval. This means that as the confidence level increases, the interval becomes wider to encompass a larger range of values, thereby increasing the likelihood that the true parameter is included. Conversely, a lower confidence level results in a narrower interval, which suggests a more precise estimate but with less certainty that the interval contains the true parameter.
For example, if we consider the statement provided: "A 99 percent confidence interval would be wider than a 95 percent confidence interval (for example, plus or minus 4.5 percent instead of 3.5 percent). A 90 percent confidence interval would be narrower (plus or minus 2.5 percent, for example)." This statement is conceptually correct in illustrating the relationship between confidence level and interval width. However, the actual width of the confidence interval also depends on other factors such as the sample size, the standard deviation of the population, and the desired precision of the estimate.
The formula for calculating the width of a confidence interval typically involves the standard error of the statistic, which is a measure of how much the sample statistic is expected to vary from the true population value. The margin of error, which is added and subtracted to the sample statistic to create the interval, is often calculated using the critical value from the standard normal distribution (also known as the z-score) corresponding to the desired confidence level. The higher the confidence level, the larger the critical value, and thus the wider the interval.
In summary, a 90% confidence interval is indeed narrower than a 95% confidence interval because it reflects a higher degree of precision but with less certainty. It's crucial for researchers and practitioners to choose an appropriate confidence level based on the context and the trade-off between precision and certainty they are willing to accept.
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A 99 percent confidence interval would be wider than a 95 percent confidence interval (for example, plus or minus 4.5 percent instead of 3.5 percent). A 90 percent confidence interval would be narrower (plus or minus 2.5 percent, for example).read more >>