Question

A plant manufacturers part whose lengths are normally distributed with a mean of 16.3 centimeters and a standard deviation of 2.5 centimeters. 20% of all parts are below what length?

Answer 1

20% of all parts are below 14.2 cm

Explanation:

The formula of the z score is z = (x - μ)/σ, where μ is the mean, σ is the standard deviation

∵ P(x < h) = 20%

∵ 20% = (20/100) = 0.2

∴ P(x < h) = 0.2

Let us use the normal distribution table to find z-score of the corresponding area 0.2

∵ z-score of the corresponding area 0.2 is - 0.845

∵ The mean is 16.3 cm

∴ μ = 16.3

∵ The standard deviation is 2.5 cm

∴ σ = 2.5

- Substitute these values in the formula above to find h

∵ -0.845 = (h - 16.3)/2.5

- Multiply both sides by 2.5

∴ - 2.1125 = h - 16.3

- Add 16.3 to both sides

∴ 14.1875 = h

- Round it to the nearest tenth

∴ h = 14.2 cm

20% of all parts are below 14.2 cm