When the population standard deviation is known, what z-value is used for constructing confidence intervals?

When constructing confidence intervals for a population mean where the population standard deviation is known, a specific z-value is used based on the desired level of confidence. For a 95% confidence level, which is one of the most commonly used intervals, the corresponding z-value is approximately 1.96. This value comes from the standard normal distribution, representing the point at which 95% of the area under the curve falls within +/- 1.96 standard deviations from the mean.

To further elaborate, the z-distribution assumes a normal distribution of the data and utilizes the standard deviation to determine how far from the mean we can expect the sample mean to lie. By using a z-value of 1.96, we can create an interval that has a 95% probability of containing the true population mean, enabling effective estimation in statistical analyses.

Other values mentioned are not appropriate for confidence interval construction with a known population standard deviation: 0 does not represent any meaningful confidence level; the t-value is used when the population standard deviation is unknown and sample sizes are small; "Any standard normal value" lacks the specificity required to establish an accurate confidence interval.

Thus, using 1.96 is the standard approach for constructing confidence intervals when the population standard