Final answer:
To find the z-values that bound the middle 80%, one must determine the 10th and 90th percentiles of the standard normal distribution, commonly around -1.28 and 1.28 respectively, and illustrate this on a graph with the desired area shaded.
Step-by-step explanation:
To find the two z values that encompass the middle 80% of the area under a normal distribution curve, one must find the z-scores that correspond to the 10th and 90th percentiles. Since there is 100% of the area under the entire curve, and we are trying to leave out 20% (10% on each side), we are left with the middle 80%. Using a standard normal distribution table or calculator, we look up the z-score for the area to the left being 0.10 and the area to the left being 0.90.
For the 10th percentile, we should find a z-score around -1.28, and for the 90th percentile, a z-score around 1.28. The graph would show the normal distribution curve with the middle 80% of the area shaded between these two z-scores, and 10% of the area in each tail.
The region in the tails (left tail having an area of 0.10 and right tail having an area of 0.10) represents the values that lie outside the common middle 80%. Understanding this concept is vital in creating confidence intervals and making statistical inferences.