Question

Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.06 and a standard deviation of 1.52. Using the empirical rule, what percentage of American women have shoe sizes that are greater than 11.1? Please do not round your answer.

Answer #1

In statistics, the 68–95–99.7 rule, also known as the
three-sigma rule or empirical rule, states that nearly all values
lie within three standard deviations of the mean in a normal
distribution.

About 68.27% of the values lie within one standard deviation of the
mean.

Similarly, about 95.45% of the values lie within two standard deviations of the mean.

Nearly all (99.73%) of the values lie within three standard
deviations of the mean.

where x is an observation from a normally distributed random
variable, μ is the mean of the distribution, and σ is its standard
deviation:

P( μ – σ ≤ x ≤ μ + σ ) ≈ 0.6827

P( μ –2σ ≤ x ≤ μ + 2σ ) ≈ 0.9545

P( μ –3σ ≤ x ≤ μ + 3σ ) ≈ 0.9973

Here μ = 8.06 and σ = 1.52

μ + σ = 9.58 ; μ – σ = 6.54

μ + 2σ = 11.1 ; μ – 2σ = 5.02

So, Using the empirical rule, the approximate percentage of
American women have shoe sizes that are no more then 11.1 and no
less than 5.02 = 95.45%

So, Using the empirical rule, the approximate percentage of
American women have shoe sizes that are no more then 11.1 = 95.45%
+ (100-95.45)/2 % = 97.725%

So, Using the empirical rule, the approximate percentage of American women have shoe sizes that are greater than 11.1

= 100%-97.725% =2.28%

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