Question

The Economist collects data each year on the price of a Big Mac in various countries around the world. A sample of McDonald's restaurants in Europe in May 2009 resulted in the following Big Mac prices (after conversion to U.S. dollars): \(\begin{array}{llllllll}3.80 & 5.89 & 4.92 & 3.88 & 2.65 & 5.57 & 6.39 & 3.24\end{array}\) The mean price of a Big Mac in the U.S. in May 2009 was \$3.57. For purposes of this exercise, you can assume it is reasonable to regard the sample as representative of European McDonald's restaurants. Does the sample provide convincing evidence that the mean May 2009 price of a Big Mac in Europe is greater than the reported U.S. price? Test the relevant hypotheses using \(\alpha=0.05 .\) (Hint: See Example 12.12)

Short answer

The mean May 2009 price of a Big Mac in Europe is statistically significantly greater than the reported U.S. price. The null hypothesis is rejected at a 5% level of significance.

Step-by-step solution

Null hypothesis and alternative hypothesis

Set the null hypothesis: The mean May 2009 price of a Big Mac in Europe equals the reported U.S. price, denoted as \(H_0: \mu = 3.57\). The alternative hypothesis: The mean May 2009 price of a Big Mac in Europe is greater than the reported U.S. price, represented as \(H_1: \mu > 3.57\).

Test statistic

Compute the sample mean and standard deviation. From the given prices, the sample mean \(\overline{X} ≈ 4.605\) and sample standard deviation \(s ≈ 1.25\). The test statistic \(t\) comes from the formula \(t= (\overline{X} - \mu) / (s/\sqrt{n})\), where \(\overline{X}\) is the sample mean, \(\mu\) is the population mean from null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the number of observations. Substituting the values, we get \(t ≈ 2.218\).

Calculate the critical value and p-value

The critical value for a one-tail t-test at a 0.05 alpha level with 7 degrees of freedom is approximately 1.895. The calculated test statistic \(t ≈ 2.218\) is above the critical value. The p-value can be computed using a t-distribution table or software. The p-value is the probability of getting a test statistic equal to or more extreme than that observed, assuming the null hypothesis is true. The p-value is approximately 0.0295.

Decision making and conclusion

If the p-value is less than or equal to 0.05, reject the null hypothesis. In this case, the p-value (approximately 0.0295) is less than \(\alpha = 0.05\), so we reject the null hypothesis. There is enough evidence to say that the mean May 2009 price of a Big Mac in Europe is greater than the reported U.S. price at 5% level of significance.