The z-score for the lower bound is -1.645, and the z-score for the upper bound is 1.645. By calculating the actual values using the mean and standard deviation, we found that the interval is approximately 74.83 to 125.17.
The shortest interval that covers 90% of WISC scores can be found using the concept of z-scores. A z-score measures how many standard deviations a particular value is from the mean.
Given that the mean (μ) of the WISC scores is 100 and the standard deviation (σ) is 15, we can calculate the z-score corresponding to the lower and upper bounds of the interval that covers 90% of the scores.
To find the z-score for the lower bound, we need to find the z-score that corresponds to the 5th percentile. The 5th percentile is the point below which 5% of the scores fall. We can use a z-table or calculator to find that the z-score for the 5th percentile is approximately -1.645.
To find the z-score for the upper bound, we need to find the z-score that corresponds to the 95th percentile. The 95th percentile is the point below which 95% of the scores fall. Again, using a z-table or calculator, we find that the z-score for the 95th percentile is approximately 1.645.
Now we can calculate the actual values for the lower and upper bounds of the interval using the formula:
Lower bound = μ + (z-score for lower bound * σ)
Upper bound = μ + (z-score for upper bound * σ)
Substituting the values, we get:
Lower bound = 100 + (-1.645 * 15) ≈ 74.83
Upper bound = 100 + (1.645 * 15) ≈ 125.17
Therefore, the shortest interval that covers 90% of the WISC scores is approximately 74.83 to 125.17.