7.2 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. 7.1 (jamovi) and Fig. 7.2 (SPSS).

Summary statistics for the diameter of Eagle Boys' large pizzas; jamovi

FIGURE 7.1: Summary statistics for the diameter of Eagle Boys’ large pizzas; jamovi

Summary statistics for the diameter of Eagle Boys' large pizzas; SPSS, slightly edited

FIGURE 7.2: Summary statistics for the diameter of Eagle Boys’ large pizzas; SPSS, slightly edited

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

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

  3. Write down the values of \(\sigma\) and \(s\).

  4. Compute the value of the standard error of the mean.

  5. Explain the difference in meaning between \(s\) and \(\text{s.e.}(\bar{x})\) here.

  6. 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?

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

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

  9. Write a statement that communicates your 95% CI for the mean pizza diameter.

  10. What are the statistical validity conditions?

  11. Which of these conditions must we assume are met for this CI to be statistically valid? Is it necessary to use Fig. 7.3? 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.
Histogram for the diameter of Eagle Boys' large pizzas

FIGURE 7.3: Histogram for the diameter of Eagle Boys’ large pizzas

  1. If we wanted to estimate the population mean diameter to within 1mm (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?
  2. 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.


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References

Dunn, Peter K. 2012. “Assessing Claims Made by a Pizza Chain.” Journal of Statistical Education 20 (1). www.amstat.org/publications/jse/v20n1/dunn.pdf.