Heights of women have a bellshaped distribution with a mean

Heights of women have a bell-shaped distribution with a mean of 161 cm and a standard deviation of 6 cm. Using Chebyshev\'s theorem, what do we know about the percentage of women with heights that are within 3 standard deviations of the mean? What are the minimum and maximum heights that are within 3 standard deviations of the mean?

At least _____% of women have heights within 3 standard deviations of 161 cm.

(Round to the nearest percent as needed.)

Solution

Minimum and maximum heights within the 3 standard deviations of the mean are:

161 - 3(6) = 161 - 18 = 143 cm

161 + 3(6) = 161 + 18 = 179 cm

By chebyshev\'s theorem,

where k = 3,

The percentage that will lie within +/- 3 standard deviations is:

1 - 1/k² = 1 - 1 / 3²

= 1 - 1/9

= 8/9

= 88.88 % of the values or 89% of the values.

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