According to the National Center for Health Statistics, the mean height of an American male is 69.3 inches and the mean height of an American female is 63.8 inches. The standard deviation for both genders is 2.7 inches.
According to Chebyshev’s Theorem 75% of the data for your gender lies between what two heights?
If height is assumed to be normal, what percentage of the data lies between those same two heights
To find the range of data according to Chebyshev's Theorem, we need to consider the percentage within a certain number of standard deviations from the mean.
According to Chebyshev's Theorem, at least (1 - 1/k^2) percent of the data lies within k standard deviations from the mean, where k is any value greater than 1.
In this case, we want to find the range for which 75% of the data lies within. Therefore, we need to find the value of k that satisfies (1 - 1/k^2) = 0.75.
Solving this equation, we have 1/k^2 = 0.25, which implies k^2 = 4, and k = 2.
So, according to Chebyshev's Theorem, at least 75% of the data lies within 2 standard deviations from the mean.
To find the range of heights for males, we can calculate the lower and upper bounds using the mean and standard deviation:
Lower bound = mean - (2 * standard deviation)
Upper bound = mean + (2 * standard deviation)
For males:
Lower bound = 69.3 - (2 * 2.7) = 63.9 inches
Upper bound = 69.3 + (2 * 2.7) = 74.1 inches
Therefore, according to Chebyshev's Theorem, 75% of the data for males lies between 63.9 inches and 74.1 inches.
Now, to find the percentage of data that lies within the same range if height is assumed to be normal, we can use the properties of the normal distribution.
Since the normal distribution is symmetric, if a certain percentage lies below a certain value, the same percentage lies above the equivalent value.
If 75% of the data lies between 63.9 inches and 74.1 inches, we can subtract the percentage that lies below 63.9 inches from 100% to find the percentage within the desired range.
To find the percentage of data below 63.9 inches for a normal distribution, we calculate the z-score using the formula:
z = (x - mean) / standard deviation
For males:
z = (63.9 - 69.3) / 2.7 = -1.99
Using a standard normal distribution table or a calculator, we find that the cumulative probability (area under the curve) to the left of -1.99 is approximately 0.025.
Therefore, the percentage of data between 63.9 inches and 74.1 inches for males, assuming a normal distribution, is approximately (100% - 2.5% - 2.5%) = 95%.