To find the z-score that separates the bottom 90% from the top 10% in a standard normal distribution, we need to locate the value that corresponds to the cumulative probability of 0.90.
The standard normal distribution is characterized by a mean of 0 and a standard deviation of 1. The z-score is a statistical measurement that describes a value’s relationship to the mean of a group of values. A z-score of 0 indicates the value is identical to the mean, while a positive z-score indicates a value above the mean, and a negative z-score represents a value below the mean.
To find the z-score that corresponds to the bottom 90%, we can use a z-table, which provides the cumulative probabilities associated with z-scores. We can also use statistical software or a calculator that provides the inverse of the cumulative distribution function (CDF).
1. **Using a Z-table**: Look for the value closest to 0.90 in the body of the z-table. The closest value is usually around 0.8997 or 0.9000, which corresponds to a z-score of approximately 1.28.
2. **Using Statistical Software/Calculator**: If using software, you can input the cumulative probability of 0.90 to get the z-score directly. Most calculators will yield a value close to 1.2816.
Therefore, the z-score that separates the bottom 90% from the top 10% is approximately 1.28 or 1.2816. This means that around 90% of the data under the standard normal curve falls below this z-score.