Question

If for three individuals the obtained z-scores were 3.00; -2.00; and 1.64, what percentage of individuals scored higher than them, respectively, according to the z-table?

Answer 1

To find out how many individuals scored higher than the given z-scores of 3.00, -2.00, and 1.64, a z-table is used to show the cumulative probability left of a z-score. The percentages of individuals scoring higher are approximately 0.13%, 97.72%, and 5.05%, respectively.

Step-by-step explanation:

The question asks for the percentage of individuals who scored higher than the given z-scores on a standard normal distribution. To find these values, one would typically refer to a z-table which shows the cumulative probability (or area) up to a given z-score.

For Each Z-Score:

1. A z-score of 3.00: Looking at a z-table, we see that the area to the left of a z-score of 3.00 is approximately 0.9987. This means that about 99.87% of scores are lower than this, implying that only (1 - 0.9987) × 100% ≈ 0.13% have scored higher.
2. A z-score of -2.00: The area to the left of a z-score of -2.00 is 0.0228, indicating that roughly 2.28% of scores are lower, so approximately 97.72% have scored higher.
3. A z-score of 1.64: The area to the left is approximately 0.9495. Hence, (1 - 0.9495) × 100% ≈ 5.05% of individuals scored higher.

Using the empirical rule, which is also known as the 68-95-99.7 rule, we could estimate these percentages. However, the z-table offers more precise values.