A 95 percent confidence interval with lower and upper limits of 1.3886 and 2.5314 suggests that there is a 95 percent chance that the true population parameter is between those two values. The width of the interval is influenced by the z-score associated with the confidence level, and a higher confidence level would result in a wider interval to ensure greater certainty.
The student has provided the lower and upper limits of a confidence interval with an error of 1.96, which typically corresponds to a 95 percent confidence interval. The lower limit is 1.3886 and the upper limit is 2.5314. To reach a conclusion from these values, we must understand that the 95 percent confidence interval is an estimate of where the true population parameter (like a mean or proportion) lies, with 95 percent certainty. In this context, it means that there is a 95 percent chance that the true population parameter is between 1.3886 and 2.5314. If the interval does not include a value of interest (like zero or another number), we can be 95 percent confident that the true parameter is different from this value.
The concept of confidence intervals is crucial in statistics because it provides a range of plausible values for the parameter being estimated, rather than just a single point estimate. A wider interval implies greater uncertainty, whereas a narrower interval indicates more precision. When the confidence level is increased, the confidence interval becomes wider to account for the increased certainty required to capture the true population parameter. Conversely, decreasing the confidence level results in a narrower interval. The error of 1.96 used here corresponds to the standard error multiplied by the z-score for a 95 percent confidence interval, which reflects the number of standard errors away from the mean a corresponding value lies.