## 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.'

1. What is Jake's misunderstanding?

2. 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?

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 $$\mu$$?