8.2 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 \(44\) 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 \(\mu\)?