I am working with standard deviation and the empirical rule.
The problem has the heights of basketball players (women) and we first had to turn their heights into feet rounded to hundredths. (eg. 5'10' = 5.83)
The mean came out to be 5.92 which is equivalent to a 5'11" woman and the standard deviation is .22.
I know the next one up is 6.14, 6.36, 6.58, and down is 5.70, 5.48, and 5.26, but how do I equate those to heights?
Such as, if 5.92, the mean is a 5'11" person,, what is the height of a person at 6.14? (I know the answer is 6'0", but do not know how that was obtained)
if std is .22, the 6.14 is 5.92 + 0.22 std
So find that in the Z table
Yes, I understand that -- I need to write the heights back in feet and inches
To determine the height of a person at a given value that is one standard deviation above the mean, you can use the z-score formula. The z-score measures the number of standard deviations a value is above or below the mean.
The formula for finding the z-score is:
z = (x - μ) / σ
Where:
x is the given value
μ is the mean
σ is the standard deviation
In this case, the mean (μ) is 5.92 and the standard deviation (σ) is 0.22.
To find the height of a person at 6.14, you can use the formula as follows:
z = (6.14 - 5.92) / 0.22
Now, you can calculate:
z = 0.22 / 0.22
Simplifying:
z = 1
Since the z-score of 1 corresponds to the value one standard deviation above the mean, you can use the empirical rule to find the height.
According to the empirical rule, for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean.
Therefore, if the mean height is equivalent to a 5'11" person, you would add one inch to get the new height. Thus, a person at 6.14 would be approximately 6'0" tall.
To determine the height of a person at a given z-score using the empirical rule, you need to know the mean and standard deviation of the data set. In this case, the mean is 5.92 (equivalent to 5'11") and the standard deviation is 0.22.
The empirical rule 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.
To calculate the height of a person at a specific z-score, you need to convert the z-score into a number of standard deviations from the mean.
To find the z-score for a given height, you can use the formula:
z = (x - mean) / standard deviation
Let's use the example of finding the height of a person at z = 1 (which corresponds to a z-score of 1).
To find the height at z = 1:
z = (x - mean) / standard deviation
1 = (x - 5.92) / 0.22
Now, you need to solve this equation for x (the height). Rearranging the equation, we have:
x - 5.92 = 0.22
x = 5.92 + (0.22 * 1)
x = 5.92 + 0.22
x = 6.14
So, a z-score of 1 corresponds to a height of 6.14 (equivalent to 6'0").