7.3 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."
Identify the two concepts that Jake is confusing. How would you clarify Jake's confusion?
Then, Jake and Claire 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 says, "A mean of \(44.9\) matches per box? You can't have \(0.9\) of a match! That's dumb." How would you respond?
"The claim is \(45\) matches per box on average, but the mean really is \(44.9\)!" said Jake. "They're liars! Liar, liar, pants on..."
Identify the two concepts that Jake is confusing. How would you clarify Jake's confusion?
What two broad reasons could explain why the sample mean is not \(45\)?
Claire says, "The mean won't be exactly \(45\) in every sample of \(25\) boxes! Let's work out the confidence interval."
Why does Claire think a CI is needed? What will it tell them?
Compute 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! I'm calling my lawyer!"Is Jake correct that the manufacturer must be lying? 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 \(\mu\)?