91Ó°ÊÓ

The Economist collects data each year on the price of a Big Mac in various countries around the world. The price of a Big Mac for a sample of McDonald's restaurants in Europe in May 2009 resulted in the following Big Mac prices (after conversion to U.S. dollars): \(\begin{array}{llllll}3.80 & 5.89 & 4.92 & 3.88 & 2.65 & 5.57\end{array}\) \(\begin{array}{ll}6.39 & 3.24\end{array}\) The mean price of a Big Mac in the U.S. in May 2009 was \(\$ 3.57\). For purposes of this exercise, 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 \(\alpha=.05\).

Step by step solution

01

Set up the Hypotheses

First, the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)) need to be established. For this exercise: \(H_0: \mu = \$3.57 - this signifies that the mean price in Europe equals the U.S. mean price. \) \(H_1: \mu > \$3.57 - Represents that the mean price in Europe is greater than the U.S. mean price.\)

02

Calculate Sample Mean and Sample Standard Deviation

Add up all the given European Big Mac prices and divide by the total number of observations to get the sample mean (\(\overline{x}\)). The sample standard deviation (s) can be obtained by calculating the square root of the variance, which itself is the average of the squared deviations from the mean. For this data set: \(\overline{x} = (\$3.80+\$5.89+\$4.92+\$3.88+\$2.65+\$5.57+\$6.39+\$3.24)/8 = \$4.505 \) For standard deviation, use a standard deviation calculator inputting all the European data set values to get \(s = \$1.21\)

03

Conduct the Hypothesis Test

Since the sample size is smaller than 30, therefore a t-test will be used. The test statistic, t, can be calculated as: \(t = (\overline{x} - \mu_{0}) / (s / \sqrt{n}) \) Where \(n\) is the sample size, \(\mu_{0}\) is the population mean, in this case the U.S. mean price, \(\$3.57\). Calculation yields: \(t = (\$4.505 - \$3.57) / (\$1.21 / \sqrt{8}) = 2.44 \) Then, compare this calculated t-value to the t critical value from the t-Distribution table given \(\alpha = 0.05\) and degrees of freedom = \(n - 1 = 7\). If the calculated t-value is larger than the standard t-value (one-tailed test), reject the null hypothesis.

04

Make a Conclusion

If the calculated t-value is higher than the table value, we reject the null hypothesis and conclude that the mean price of a Big Mac in Europe is significantly larger than in the U.S. On the other hand, if the table value is higher, we fail to reject the null hypothesis, suggesting there's not enough evidence to support that the mean price of the Big Mac in Europe is larger than that in the U.S. We would have to look up the t-table to finalize the conclusion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

Understanding the null hypothesis is crucial for hypothesis testing. It's a statement suggesting that there is no effect or no difference, and it serves as a starting point for statistical analysis. In our Big Mac price scenario, the null hypothesis (\( H_0 \)) posits that the mean price of a Big Mac in Europe is equal to the mean price in the United States, specifically \( \$3.57 \). This is the benchmark against which we test the possibility of a difference in pricing.

Alternative Hypothesis

Complementary to the null hypothesis is the alternative hypothesis (\( H_1 \) or \( H_a \)), which represents what we aim to support with our data. For the Big Mac exercise, the alternative hypothesis claims that the average price in Europe is greater than in the U.S., or \( \mu > \$3.57 \). Demonstrating the alternative hypothesis typically requires showing that the observed difference is unlikely to have occurred due to chance if the null hypothesis were true.

T-Test

When we have a smaller sample size (less than 30), or we don’t know the population standard deviation, we use the t-test. It's a statistical tool to determine if there is a significant difference between the means of two groups. In our case, we are using a one-sample t-test to compare the sample mean of the Big Mac prices in Europe with the known U.S. mean price.

Sample Mean

The sample mean (\( \overline{x} \) is the average value of our sample data. It’s calculated by adding up all the sample values and dividing by the number of observations. For our Big Mac price data, the sample mean calculated from the sample of European prices is \( \$4.505 \), which we use to compare against the U.S. mean price.

Sample Standard Deviation

Sample standard deviation (s) measures the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. Our calculated sample standard deviation for European Big Mac prices is \( \$1.21 \) and reflects the variability within the sample prices.

Degrees of Freedom

Degrees of freedom (df) are an essential concept in statistical tests, including the t-test. They represent the number of values in the final calculation of a statistic that are free to vary. In the context of the t-test, the degrees of freedom is the sample size minus one (\( n - 1 \)). For our set of Big Mac prices, we have a sample size of 8, so the degrees of freedom we use is 7.

Statistical Significance

Statistical significance is a determination about the null hypothesis. In hypothesis testing, we use a significance level (\( \alpha \)), typically set at 0.05 or 5%, to decide whether to reject the null hypothesis. It represents the probability threshold under which we determine our test results are significant. If our test statistic is past this threshold, we have grounds to reject the null hypothesis, suggesting our findings are not due to random chance.

P-Value

The p-value is a metric used to summarize the strength of the evidence against the null hypothesis provided by our sample. It represents the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. A small p-value (\( p < \alpha \) indicates strong evidence against the null hypothesis, so it is rejected, and the alternative hypothesis is accepted. In our example, we would need to compute or look up the p-value associated with the t-statistic to decide on the hypothesis.