Heights of women have a bell-shaped distribution with a mean of 161 cm and a standard deviation of 7 cm. Using Chebyshev’s theorem, what do we know about the percentage of women with heights that are within 2 standard deviations of the mean? What are the minimum and maximum heights that are within 2 standard deviations of the mean?
To determine what we know about the percentage of women with heights within two standard deviations of the mean using Chebyshev's theorem, we need to understand Chebyshev's bound.
Chebyshev's theorem states that for any data set, regardless of the shape of its distribution, a minimum proportion of the data must fall within a certain number of standard deviations of the mean. Specifically, for any k greater than 1, at least 1 - 1/k^2 of the data will fall within k standard deviations of the mean.
In this case, we want to find the percentage of women with heights within 2 standard deviations of the mean. Using Chebyshev's theorem, we set k = 2.
According to Chebyshev's theorem, at least 1 - 1/k^2 = 1 - 1/2^2 = 1 - 1/4 = 3/4 = 75% of the data will fall within 2 standard deviations of the mean.
Hence, we can conclude that at least 75% of women have heights within 2 standard deviations of the mean.
To calculate the minimum and maximum heights within 2 standard deviations of the mean, we need to use the formula:
Minimum Height = Mean - (Standard Deviation * Number of Standard Deviations)
Maximum Height = Mean + (Standard Deviation * Number of Standard Deviations)
In this case, the mean is 161 cm, and the standard deviation is 7 cm. We want to find the range within 2 standard deviations, so the number of standard deviations is 2.
Minimum Height = 161 cm - (7 cm * 2) = 161 cm - 14 cm = 147 cm
Maximum Height = 161 cm + (7 cm * 2) = 161 cm + 14 cm = 175 cm
Therefore, the minimum height within 2 standard deviations of the mean is 147 cm, and the maximum height is 175 cm.
According to Chebyshev's theorem, regardless of the shape of the distribution, at least (1 - 1/k^2) of the data will fall within k standard deviations of the mean, where k is any positive number greater than 1.
In this case, since we are interested in the percentage of women with heights that are within 2 standard deviations of the mean, k will be equal to 2.
Applying the formula, we have:
Percentage within 2 standard deviations = 1 - 1/k^2
Percentage within 2 standard deviations = 1 - 1/2^2
Percentage within 2 standard deviations = 1 - 1/4
Percentage within 2 standard deviations = 1 - 0.25
Percentage within 2 standard deviations = 0.75
Therefore, we can conclude that at least 75% of the women's heights will fall within 2 standard deviations of the mean.
To calculate the minimum and maximum heights within 2 standard deviations of the mean, we need to consider the following:
Minimum height within 2 standard deviations = mean - (2 * standard deviation)
Minimum height within 2 standard deviations = 161 cm - (2 * 7 cm)
Minimum height within 2 standard deviations = 161 cm - 14 cm
Minimum height within 2 standard deviations = 147 cm
Maximum height within 2 standard deviations = mean + (2 * standard deviation)
Maximum height within 2 standard deviations = 161 cm + (2 * 7 cm)
Maximum height within 2 standard deviations = 161 cm + 14 cm
Maximum height within 2 standard deviations = 175 cm
Therefore, the minimum height within 2 standard deviations of the mean is 147 cm, and the maximum height within 2 standard deviations of the mean is 175 cm.
Chebyshev's Theorem says:
1. Within two standard deviations of the mean, you will find at least 75% of the data.
2. Within three standard deviations of the mean, you will find at least 89% of the data.
Here's how the formula shows this:
Formula is 1 - (1/k^2) ---> ^2 means squared.
If k = 2 (representing two standard deviations), we have this:
1 - (1/2^2) = 1 - (1/4) = 3/4 or .75 or 75%
If k = 3 (representing three standard deviations), we have this:
1 - (1/3^2) = 1 - (1/9) = 8/9 or approximately .89 or 89%
I'll let you take it from here.