7.1 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 package containing twenty boxes of matches, and Jake counted the number of matches in one of the 20 boxes. There was 44 matches.

‘Oh wow. Just wow.’ said Jake. ‘They lie. There’s only 44 in this box.’

  1. What is Jake’s misunderstanding?

  2. Then, they counted the number of matches in each of the twenty boxes.

    Claire found that the mean number of matches per box was 44.9 matches, and the standard deviation was \(0.11\).

    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?

  3. ‘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?

  4. What two broad reasons could explain why the sample mean is not 45?

  5. ‘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?

  6. What is an approximate 95% confidence interval for the mean for Claire’s sample?

  7. ‘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?

  8. In this scenario, what does \(\bar{x}\) represent? What is the value of \(\bar{x}\)?

  9. In this scenario, what does \(\mu\) represent? What is the value of \(\mu\)?