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

99,7 %

Step-by-step 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