Question

Suppose a histogram for the heights of two-year-old children is approx bell-shaped as shown below, with a mean of 27 inches and a standard deviation of 1.5 inches. Using the 68-95-99.7 rule, about what proportion of heights are between 25.5 and 28.5?

Answer 1

Suppose a histogram for the heights of two-year-old children is approx bell-shaped.

Le X be the random variable that denotes the heights of two-year-old children.

Therefore, X approximately follows the normal distribution.

Given, the mean = 27 inches and the standard deviation = 1.5 inches.

According to the 68-95-99.7 rule, 68%, 95% and 99.7% of the heights are within 1, 2 and 3 standard deviations of the mean respectively i.e

P( - < X < + ) = 0.68

P( - 2 < X < + 2) = 0.95

P( - 3 < X < + 3) = 0.997

Using the 68-95-99.7 rule for this question,

P(25.5 < X < 28.5) = P(27 - 1.5 < X < 27 + 1.5)

                               = P( - < X < + )

                               = 0.68

Therefore, 68% of heights are between 25.5 and 28.5 inches.