A bottling manufacturing fills soda bottles with a label that says content is 12

Question

A bottling manufacturing fills soda bottles with a label that says content is 12 ounces. Quality control studies have shown that the bottle filling process is normally distributed with a mean equal to 11.7 ounces and a standard deviation equal to 0.35 ounces.

A) What is the probability the content of the bottle is less than 11 ounces?

B) what is the probability the content of the bottle meets what the label promises.

C) What is the probability the content of a bottle is between 10.2 to 12.5 ounces?

D) The manager wants to know the amount of liquid in the top 10% of the most-filled bottles. What amount of liquid content represents the cut-off for the top 10% of the bottles?

Explanation / Answer

Mean is 11.7 and s is 0.35

z is calculated as (x-mean)/s

A) P(x<11) =P(z<(11-11.7)/0.35)=P(z<-2) or 1-P(z<2), from normal distribution table it gives 1-0.9772=0.0228

B) P(x<12)=P(z<(12-11.7)/0.35)=P(z<0.86) , from normal table we get 0.8051

C) P(10.2<x<12.5) =P((10.5-11.7)/0.35<z<(12.5-11.7)/0.35)=P(-3.43<z<2.29) or P(z<2.29)-P(z<-3.43)

or P(z<2.29)-(1-P(z<3.43))=0.9890-(1-1)=0.9890

D) We need to look at value which is 90% above lowest value. i.e look for z for 0.9 area we get 1.28

thus answer is mean+z*s =11.7+1.28*0.35=12.148

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