Find the z scores that cut-off the most extreme 6% under the standard normal curve. a. ±1.55 b. ±1.88 c. ±0.52 d. ±0.51
The correct answer and explanation is:
The correct answer is: b. ±1.88
Explanation (Approximately 300 Words):
In statistics, a z-score represents how many standard deviations a data point is from the mean of a standard normal distribution, which has a mean of 0 and a standard deviation of 1. The “most extreme 6%” refers to the outer 6% of the distribution — that is, 3% in each tail (both left and right) of the standard normal curve.
To find the z-scores that cut off the most extreme 6%, we look for the values where 3% of the area lies in the lower tail (left side) and 3% lies in the upper tail (right side). This means we want to find the z-scores that leave 94% (100% – 6%) of the data in the center of the curve, so 3% lies below the lower bound and 3% lies above the upper bound.
Steps:
- We want to find the z-scores such that the area to the left is 0.03 for the lower bound and 0.97 (1 – 0.03) for the upper bound.
- Using a z-table or statistical software, we look for the z-score corresponding to 0.03 cumulative probability.
- The z-score for 0.03 is approximately -1.88.
- Because the normal distribution is symmetric, the corresponding upper z-score is +1.88.
So, the two z-scores that cut off the most extreme 6% (3% in each tail) are approximately ±1.88.
Summary:
- Total extreme area: 6%
- Each tail: 3%
- Central area: 94%
- Corresponding z-scores: ±1.88
These values represent the cut-off points beyond which lie the most unusual 6% of observations in a standard normal distribution — often used in hypothesis testing and confidence interval calculations.