EXAMPLE 8 Finding two symmetric sample mean values using the Central Limit Theorem for Means

Refer to Example 7. If we take samples of size $n=36$, find the two symmetric values of $\overline{x}$ that contain the middle 95% of sample mean city mpg values.

Solution

Here, we use the instructions from the Finding Percentiles for Any Normal Distribution portion of the Section 6.5 Step-by-Step Technology Guide on page 380. We have ${\mu }_{\overline{x}}=20$ and ${\sigma }_{\overline{x}}=1$. We want the middle 95% of sample means, so we ask the TI-83/84 calculator to give us the sample mean with 2.5% to the left of it and the sample mean with 97.5% to the left of it, as shown in Figure 10. We get ${\overline{x}}_1\approx 18.04$ as the 2.5th percentile of sample means and ${\overline{x}}_2\approx 21.96$ as the 97.5th percentile of sample means. These are the two symmetric values of the sample mean that contain the middle 95% of all sample mean city mpg values.