Question

The distribution of the heights of the first grade students is mound – shaped and symmetric with the mean height of 140 cm and standard deviation of 5 cm.
According Empirical rule

What % of heights is between 125 and 155 cm

What % of heights is between 135 and 150 cm

What % of heights is less than 130 cm

What % of heights is less than 130 cm

Answer 1

Solution :

According to empirical rule of normal distribution about 68% of all possible observations of a normal distribution lie between (μ - σ) and (μ + σ), about 95% of all possible observations of a normal distribution lie between (μ - 2σ) and (μ + 2σ) and about 99.7% of all possible observations of a normal distribution lie between (μ - 3σ) and (μ + 3σ).

We have, μ = 140 cm and σ = 5 cm

1) (μ - 3σ) = (140 - 5 × 3) = 125

(μ + 3σ) = (140 + 5 × 3) = 155

Hence, 99.7% of heights is between 125 and 155 cm.

2) (μ - σ) = (140 - 5 ) = 135

(μ + 2σ) = (140 + 5 × 2) = 150

About 34% of observations lie between (μ - σ) and μ. And 47.5% of observations lie between μ and (μ + 2σ).

Hence, 34% of heights lies between 135 and 140 and 47.5% of heights lie between 140 and 150.

Hence, total 81.5% of heights lie between 135 cm and 150 cm.

3) (μ - 2σ) = (140 - 2 × 5) = 130

About 2.28% of data are less than (μ - 2σ).

Hence, 2.28% of heights are less than 130 cm.