Since 99.7% of students fall within 27 SDs of the mean, and 68% within 1 SD, adding those parts gives **≈95%** between z-scores 0 and 27 in a normal height distribution correct answer is 95%.
Understand the normal distribution and z-scores.
A normal distribution is a bell-shaped curve where most data points cluster around the mean, and fewer data points fall further away.
A z-score tells you how many standard deviations a specific data point is away from the mean. Positive z-scores indicate values above the mean, and negative z-scores indicate values below the mean.
Analyze the given information.
We know the heights of all students are normally distributed.
We want to find the percentage of students with z-scores between 0 and 27.
Use the properties of the normal distribution.
We know that 68% of the data points in a normal distribution fall within 1 standard deviation of the mean .
Each additional standard deviation encompasses a progressively smaller percentage of the data.
Calculate the area beyond 27 standard deviations.
Since 27 is much larger than 3 standard deviations (27/3 ≈ 9), we can assume essentially all data points (over 99.7%) fall within 27 standard deviations of the mean.
Combine the areas to find the final percentage.
Add the percentage of data within 1 standard deviation (68%) to the area beyond 27 standard deviations:
68% (within 1 SD) + 27% (beyond 27 SD) = 95%
Therefore, approximately 95% of all students will have z-scores between 0 and 27 in a normally distributed height dataset.