CI for Mean (σ Known)

Statistics · Axiom Academy

Constructing Confidence Intervals for Population Mean with Known Standard Deviation 1. Understanding Confidence Intervals A confidence interval gives us a range of values that likely contains the true population mean. Instead of using a single point estimate (x̄), we create an interval around it to account for sampling variability. The animation above shows how sample means from different samples vary around the true population mean. A 95% confidence interval is constructed to capture the population mean in 95% of all possible samples. 2. The Confidence Interval Formula When σ is known, we use the z-distribution to construct the confidence interval: x̄ = sample mean (center of our interval) z* = critical value from standard normal distribution σ = population standard deviation (known) σ/√n = standard error of the mean 3. Finding the Critical Value (z*) The critical value z* depends on our desired confidence level. Common confidence levels and their z* values: Higher confidence levels require wider intervals. The animation shows how the critical value relates to the area under the standard normal curve. A researcher measures the heights of 36 adults. The sample mean is x̄ = 170 cm, and the population standard deviation is known to be σ = 12 cm. Find a 95% confidence interval for the true mean height. Step 1: Identify values: x̄ = 170, σ = 12, n = 36, z* = 1.96 (for 95%) Step 2: Calculate standard error: σ/√n = 12/√36 = 12/6 = 2

This is the written version of the interactive lesson above. See the full Statistics course.