## 7.4 CIs for one mean

In 2011, *Eagle Boy's Pizza* ran a campaign that claimed (among many other claims) that Eagle Boy's pizzas were 'Real size \(12\)-inch large pizzas' in an effort to out-market *Domino's Pizza*.

Eagle Boy's made the data behind the campaign publicly available (Dunn 2012). A summary of the diameters of a sample of \(125\) of Eagle Boys' large pizzas is shown in Fig. 7.2 (jamovi) and Fig. 7.3 (SPSS).

What do \(\mu\) and \(\bar{x}\) represent in this context?

Write down the

*values*of \(\mu\) and \(\bar{x}\).Write down the

*values*of \(\sigma\) and \(s\).Compute the value of the standard error of the mean.

Explain the difference in

*meaning*between \(s\) and \(\text{s.e.}(\bar{x})\) here.If someone else takes a sample of \(125\) Eagle Boy's pizzas, will the sample mean be \(11.486\) inches again (as it is in this sample)? Why or why not?

Draw a picture of the approximate sampling distribution for \(\bar{x}\).

Compute an approximate 95% confidence interval for the mean pizza diameter.

Write a statement that communicates your \(95\)% CI for the mean pizza diameter.

What are the

**statistical**validity conditions?Which of these conditions must we

**assume**are met for this CI to be**statistically**valid? Is it necessary to use Fig. 7.4? Explain.- The sample size is greater than about \(25\).
- The population has a normal distribution.
- The population standard deviation is known.
- The sample has a normal distribution.

- If we wanted to estimate the population mean diameter to within \(1\) mm (or \(0.04\) inches) with \(95\)% confidence, what size sample would we need?

What is a reasonable level of accuracy with which we could measure the diameter of a pizza? - Do you think that, on average, the pizzas do have a mean diameter of \(12\) inches in the population, as Eagle Boy's claim? Explain.