Find the Z-scores for which 70% of the distribution’s area lies between ~Z and Z_ 0.7 1.036 -0.036 0.35
The Correct Answer and Explanation is:
To find the z-scores for which 70% of the distribution’s area lies between −Z-Z−Z and ZZZ, you need to understand that the area between these two z-scores represents the middle 70% of the normal distribution. This means that 15% of the total area lies in each of the two tails beyond these z-scores (because 100% – 70% = 30%, and this is split between the two tails).
Step 1: Determine the cumulative area.
Since the total area between the two z-scores is 70%, the cumulative area to the left of ZZZ is 0.85 (because 70% of the area is between −Z-Z−Z and ZZZ, and half of the remaining 30% is in the left tail). Therefore, the cumulative area to the left of the positive z-score is 0.85. The cumulative area to the left of the negative z-score is 0.15 (because it’s symmetric).
Step 2: Find the corresponding z-scores.
You can use a standard normal distribution table or a Z-table to find the z-scores corresponding to these cumulative areas.
- The z-score for a cumulative area of 0.85 is approximately 1.036.
- The z-score for a cumulative area of 0.15 is approximately -1.036.
Step 3: Conclusion.
Thus, the z-scores for which 70% of the distribution’s area lies between them are approximately −1.036-1.036−1.036 and 1.0361.0361.036.
These z-scores are symmetric because the normal distribution is symmetric, and they represent the points beyond which 15% of the area lies in each tail. This means that between −1.036-1.036−1.036 and 1.0361.0361.036, you capture 70% of the total area under the normal curve.
