Find the following percentiles for the standard normal distribution. Interpolate where appropriate. (Round your answers to two decimal places.) in USE SALT (a) 91st (b) 9th (c) 74th (d) 26th (e) 16th You may need to use the appropriate table in the Appendix of Tables to answer this question.
The Correct Answer and Explanation is:
To find the percentiles for the standard normal distribution, we use the z-scores that correspond to the given percentiles. These z-scores can be found using a standard normal distribution (Z) table or statistical software.
Answers:
(a) 91st percentile:
This means 91% of the data falls below this point.
Z ≈ 1.34
(b) 9th percentile:
This means 9% of the data falls below this point.
Z ≈ -1.34
(c) 74th percentile:
This means 74% of the data falls below this point.
Z ≈ 0.64
(d) 26th percentile:
This means 26% of the data falls below this point.
Z ≈ -0.64
(e) 16th percentile:
This means 16% of the data falls below this point.
Z ≈ -0.99
Explanation
In a standard normal distribution, the data is symmetrically distributed with a mean (μ) of 0 and a standard deviation (σ) of 1. The percentiles indicate the proportion of data that falls below a given z-score.
To find the z-score corresponding to a specific percentile, we use the inverse of the cumulative distribution function (CDF) for the standard normal distribution, often denoted as z = Φ⁻¹(p), where p is the percentile as a decimal.
For instance:
- For the 91st percentile, we look for the z-score such that 91% of values fall below it. Using a Z-table or calculator, we find z ≈ 1.34.
- For the 9th percentile, we seek the z-score where only 9% of values are below. This is symmetric to the 91st percentile but in the negative direction: z ≈ -1.34.
- The 74th percentile yields z ≈ 0.64, while the 26th percentile is z ≈ -0.64.
- The 16th percentile, which is commonly associated with lower outliers in data, corresponds to z ≈ -0.99.
These values are approximate and might require interpolation if using a Z-table. However, most calculators or statistical software (like Excel, Python, or SALT) can provide accurate values instantly using built-in functions like NORM.S.INV(percentile).
Understanding these percentiles is crucial in statistics, especially in quality control, test scoring, and probability estimations in real-world scenarios.
