Question

During its manufacturing process, Fantra fills its 20 fl oz bottles using an automated filling machine. This machine is not perfect and will not always fill each bottle with exactly 20 fl oz of soft drink. The amount of soft drink poured into each bottle follows a normal distribution with mean 20 fl oz and standard deviation 0.17 fl oz. The Fantra quality testing department has just carried out a routine check on the average amount of soft drink poured into each bottle. A sample of 25 bottles were randomly selected and the amount of soft drink in each bottle was measured. The mean amount of soft drink in each bottle was calculated to be 19.91 fl oz. The Fantra Chief Executive Officer believes that such a low mean is not possible and a mistake must have been made. Calculate the probability of obtaining a sample mean below 19.91 fl oz. Give your answer as a decimal to 4 decimal places.

Answer 1

The probability of obtaining a sample mean below 19.91 fl oz. is 0.0041

Explanation:

Consider the provided information.

The amount of soft drink poured into each bottle follows a normal distribution with mean 20 fl oz and standard deviation 0.17 fl oz.

A Sample of 25 bottles were randomly selected and the amount of soft drink in each bottle was measured.

The mean amount of soft drink in each bottle was calculated to be 19.91 fl oz.

Thus we have mean 20 fl oz
`X =19.91` σ=0.17 and n=25.

`z=(X-\mu_x)/((\sigma)/(\sqrt n))`

Substitute the respective values.

`z=\frac{19.91-20}{\frac{0.17}{\sqrt {25}}}`

`z=(-0.09)/((0.17)/(5))`

`z=(-0.09)/(0.034)`

`z=-2.6471`

According to normal distribution table:

`P(z<-2.6471)= 0.0040593`

The probability of obtaining a sample mean below 19.91 fl oz. is 0.0041