Final answer:
To find the standard deviation when the mean height is 65 inches and 72 inches represents the 84th percentile, we use the z-score formula. The 84th percentile corresponds to a z-score of around 1.0. Thus, the standard deviation is calculated as (72 - 65) / 1.0, which equals 7 inches.
Step-by-step explanation:
The student is asking a question regarding the determination of the standard deviation of a normally distributed data set, where the mean height is 65 inches and 72 inches represents the 84th percentile. To find the standard deviation, we can use the z-score formula because the percentile gives us the corresponding z-score. In a standard normal distribution, the 84th percentile corresponds to a z-score of approximately 1.0 since it is one standard deviation above the mean.
Here are the steps to find the standard deviation:
Use the z-table or standard normal distribution to find the z-score that corresponds to the 84th percentile.
Given that the z-score is approximately 1.0, we can use the formula: z = (X - mean) / standard deviation.
Substitute the known values: 1.0 = (72 - 65) / standard deviation.
Solve for the standard deviation: standard deviation = (72 - 65) / 1.0 = 7 inches.
The standard deviation for the height distribution of this high school class is 7 inches.