Question

The cost of 5 gallons of ice cream has a varianceof 36 with a mean of 36 dollars during the summer.What is the probability that the samplean would differ from the true mean by more than 0.6 dollars if a sample of 107 5-gallon pails is randomly selected? Roundyour answer to four decimal places.

Answer 1

Given:

`\begin{gathered} Variance=36 \\ mean=36 \end{gathered}`

To Determine: The samplean would differ from the true mean by more than 0.6 dollars

Solution

Please note that standard deviation is the square root of variance

`\begin{gathered} SD=√(Variance) \\ SD=Standard-deviation \\ SD=√(36)=6 \end{gathered}`
`\begin{gathered} S.E=(SD)/(√(n)) \\ S.E=Standard-Error \\ n=107 \\ S.E=(6)/(√(107))=0.5800 \end{gathered}`

Please note that Z is the number of SE(standard error away from the mean. Therefore

`\begin{gathered} Z=(0.6)/(0.5800) \\ Z=1.0345 \end{gathered}`
`P(|Z|<1.0345)`
`P(|Z|<1.0345)=1-P(Z<-1.0345)=1-0.1515=0.8485`

Hence the probability is 0.8485