11.1 Introduction
Could you design an experiment to check whether a coin is unbalanced? A coin is said to be balanced if each of its two faces is equally likely to occur when the coin is tossed. Hence, if we define [latex]p[/latex] as the proportion of times that a head occurs among an infinite number of coin tosses, it follows that [latex]p=0.5[/latex] if the coin is balanced, while [latex]p \neq 0.5[/latex] if the coin is unbalanced. Therefore, in order to test whether the coin is unbalanced, we may toss the coin [latex]n[/latex] times, record the number of heads observed, and then perform a one-proportion z test to test [latex]H_0: p=0.5[/latex] versus [latex]H_a: p \neq 0.5[/latex].
How about an experiment to test whether a die is unbalanced? A 6-sided die is considered balanced if each of its six faces is equally likely to occur when the die is rolled. Hence, define [latex]p_i[/latex] as the proportion of times that face [latex]i[/latex] occurs among an infinite number of rolls. If the die is balanced, then [latex]p_i = \frac{1}{6}[/latex], for [latex]i=1, 2, 3, 4, 5, 6[/latex]; if the die is unbalanced, then [latex]p_i \neq \frac{1}{6}[/latex] for at least one of the faces. Therefore, in order to test whether the die is unbalanced, we may roll the die [latex]n[/latex] times and compute the sample proportions [latex]\hat{p}_1[/latex] through [latex]\hat{p}_6[/latex]; there is evidence that the die is unbalanced, if any [latex]\hat{p}_i[/latex] is significantly different from [latex]\frac{1}{6}[/latex].
The question arises: how do we conduct a hypothesis test when there are six proportions of interest? The naïve approach is to perform six one-proportion z tests. However, this approach is problematic for two main reasons. First, it is time-consuming to conduct six consecutive hypothesis tests; a single test would be more efficient. Second, when several hypothesis tests are performed in succession, the overall type I error rate increases (this is called the multiple comparisons problem). The solution to these problems is to perform a single hypothesis test, with hypotheses
[latex]H_0: p_1 = p_2 = p_3 = p_4 = p_5 = p_6 = \frac{1}{6}[/latex] versus [latex]H_a: p_i \neq \frac{1}{6}[/latex] for at least one [latex]i = 1,2,3,4,5,6[/latex].
To test such hypotheses, we rely on new types of tests based on the chi-square distribution. The tests are referred to as chi-square tests.
