9.2 Tests for one mean
In 2011, Eagle Boys' Pizza ran a campaign that claimed that Eagle Boys' pizzas were 'Real size \(12\)-inch large pizzas' (Dunn 2012). Eagle Boys' made the data from the campaign publicly available.
A summary of the diameters of a sample of \(125\) of their large pizzas is shown in Fig. 9.1 (jamovi) and Fig. 9.2 (SPSS). We would like to test the company's claim, and ask the RQ:
For Eagle Boys' pizzas, is mean diameter actually \(12\) inches, or not?
FIGURE 9.1: Summary statistics for the diameter of Eagle Boys' large pizzas; jamovi
FIGURE 9.2: Summary statistics for the diameter of Eagle Boys' large pizzas; SPSS, slightly edited
- What is the parameter of interest?
- Write down the values of \(\bar{x}\) and \(s\).
- Determine the value of the standard error of the mean.
- Explain the difference in meaning between \(s\) and \(\text{s.e.}(\bar{x})\) in this context.
- Write the hypotheses to test if the mean pizza diameter is \(12\) inches.
- Is the alternative hypothesis one- or two-tailed? Why?
- Draw the normal distribution that shows how the sample mean pizza diameter would vary by chance, even if the population mean diameter was 12 inches.
- Compute the \(t\)-score for testing the hypotheses.
- What is the approximate \(P\)-value using the \(68\)--\(95\)--\(99.7\) rule?
- Write a conclusion. (The CI was found in the Teaching Week 7 tutorial.)
- Is it reasonable to assume the statistical validity conditions are satisfied? (Fig. 9.3 may, or may not, help.)
- Do you think that the pizzas do have a mean diameter of 12 inches in the population, as Eagle Boys' claim? Explain.
FIGURE 9.3: Histogram for the diameter of Eagle Boys' large pizzas
References
Dunn PK. Assessing claims made by a pizza chain. Journal of Statistical Education [Internet]. 2012;20(1). Available from: www.amstat.org/publications/jse/v20n1/dunn.pdf.