Question

A certain manufacturing procedure produces items that have a mean weight of 55 pounds and a standard deviation of 3.2 pounds. With what minimum probability can we assert that the weight of a randomly selected item produced by this procedure is between 45.4 pounds and 64.6 pounds

Answer 1

99,7 %

Explanation:

P [ 45,4 < X < 64,6 ] ???

P [ 45,4 < X < 64,6 ]

for X = 45,4

z₁ = ( X - μ₀ ) / σ

z₁ = 45,4 - 55 / 3,2

z₁ = - 9,6 / 3,2

z₁ = - 3

And fom z-table we find P [ X = 45,4 ] = 0,0013 or 0,13 %

z₂ = ( 64,6 - 55 )/ 3,2

z₂ = 3

And from z-table we find P [ X = 64,6 ] = 0,9987 or 99,87 %

Then

P [ 45,4 < X < 64,6 ] = 99,87 - 0,13 = 0,9974 or 99,74 %

Now if we look at the values:

μ₀ - 3*σ and μ₀ + 3*σ

we find 55 - 3*3,2 = 55 - 9,6 = 45,4 and 55 + 9,6 = 64,6

We know according to the empirical rule that values from μ₀ ± 3*σ will contains 99,7 % of all values