Final answer:
To find the z-scores that divide the middle 68% of a normal distribution, you use z1 = -1 and z2 = +1, which lie one standard deviation away from the mean on a bell-shaped curve. This is derived from the Empirical Rule and can be confirmed using a z-table.
Step-by-step explanation:
Finding z-scores for Middle 68% of a Normal Distribution
To find the z-scores z1 and z2 that separate the middle 68% of a normal distribution, we can use the Empirical Rule. This rule indicates that approximately 68% of the data in a normal distribution lies within one standard deviation from the mean. For a standard normal distribution, the mean is 0 and the standard deviation is 1.
Therefore, the z-scores that cut off the lower and upper 34% (making up 68% in the middle) are -1 and +1, respectively. To visualize this, imagine a bell-shaped curve with a line drawn at the mean (z=0). The area between z=-1 and z=+1 is shaded to represent the middle 68% of the distribution.
The z-table can be used to confirm these percentages by checking the area under the curve from the mean to z=-1 and from the mean to z=+1. Since the distribution is symmetric, the area to the left of z=-1 would be approximately 15.85%, and the area to the right of z=+1 is the same, with the middle 68% lying between them.
If a different normal distribution has a different mean or standard deviation, you would convert values to z-scores and then apply the same method to find the corresponding values on the original scale that separates the middle 68% of the distribution.