7.2 Concepts

Claire and Jake were wondering about the mean number of matches in a box. The boxes contain this statement:

An average of $45$ matches per box.

They purchased a carton containing $25$ boxes of matches, and Jake counted the number of matches in one of those $25$ boxes. There was $44$ matches.

'Oh wow. Just wow.' said Jake. 'They lie. There's only $44$ matches in this box.'

What is Jake's misunderstanding?
Then, they counted the number of matches in each of the twenty-five boxes. Claire found that the mean number of matches per box was $44.9$ matches, and the standard deviation was $0.124$ .

Jake notes that the mean is $44.9$ matches per box, and says: 'You can't have $0.9$ of a match. That's dumb.' How would you respond?
'Wow!' said Jake. 'The claim is $45$ matches per box on average, but the mean really is $44.9$ ! They're liars! Liar, liar, pants on...'

What is Jake's misunderstanding?
What two broad reasons could explain why the sample mean is not $45$ ?
'Come on, Jake,' said Claire. 'As if the mean will be exactly $45$ in a sample every single time. Let's work out the confidence interval.'

Why does Claire think a CI is needed? What will it tell them?
What is an approximate $95$ % confidence interval for the mean for Claire's sample?
'Aha---I told you so! They are absolutely lying! Your confidence interval doesn't even include their mean of $45$ !' said Jake. 'The manufacturer must be lying!'

Is Jake correct? Why or why not? What does the CI mean?
In this scenario, what does $\bar{x}$ represent? What is the value of $\bar{x}$ ?
In this scenario, what does $\mu$ represent? What is the value of $\mu$ ?