if heights of a group of girls are normally distributed with a mean of 66 inches . if 95% of the heights are between 63 inches and 69 inches, what is the standard deviation for this group
We know that 95% of the heights are between 63 inches and 69 inches.
This means that we can use the Empirical Rule, which states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean
- Approximately 95% of the data falls within two standard deviations of the mean
- Approximately 99.7% of the data falls within three standard deviations of the mean
Since we are given that 95% of the heights fall between 63 inches and 69 inches, we can assume that this range represents two standard deviations from the mean.
Using the formula for finding the range of values within k standard deviations of the mean:
range = mean ± k standard deviations
We can write:
66 ± 2s = 63 and 66 ± 2s = 69
Solving for s (the standard deviation):
66 - 2s = 63
-2s = -3
s = 1.5
66 + 2s = 69
2s = 3
s = 1.5
So the standard deviation for this group of girls' heights is approximately 1.5 inches.