You and your friend play dice games for fun, but one day you suspect that the die your friend uses is not fair. You will gather data and use the tools of hypothesis testing in this problem to provide a plausible answer as to whether or not the die is fair.
Your statistical model is (\{ 1,2,3,4,5,6\} , \{ \mathbf{P}_{\mathbf{p}} \} _{\mathbf{p} \in \Delta _6}). You roll the die 15 times, writing the sample as random variables X_1, \ldots , X_{15} \stackrel{iid}{\sim } \mathbf{P}_{\mathbf{p}^*} where \mathbf{p}^* is the true parameter. Your null and alternative hypothesis are, respectively,
\displaystyle H_0: \mathbf{p}^* \displaystyle = [1/6~ ~ 1/6~ ~ 1/6~ ~ 1/6~ ~ 1/6~ ~ 1/6]^ T.
\displaystyle H_1: \mathbf{p}^* \displaystyle \neq [1/6~ ~ 1/6~ ~ 1/6~ ~ 1/6~ ~ 1/6~ ~ 1/6]^ T.
Let \widehat{\mathbf{p} } denote the MLE for the true parameter \mathbf{p}^*. You use the following test statistic to test the hypotheses:
T_ n = n \sum _{j = 1}^6 \frac{(\widehat{p}_ j - \frac{1}{6} )^2}{\frac{1}{6} }.
T_ n converges to a random variable X \sim \chi _ k^2 for some integer k. What is k?
To find the value of k, we need to determine the degrees of freedom of the chi-squared distribution X. In this case, k is equal to the number of categories minus 1.
Since we have 6 categories (numbers 1 to 6), k = 6 - 1 = 5.
Therefore, the value of k is 5.
To determine the value of k, we need to understand the distribution of the test statistic T_n. Here, T_n follows a chi-squared distribution with k degrees of freedom, denoted by X ~ χ^2(k).
In this case, the test statistic T_n is defined as:
T_n = n * ∑[(∆p_j - 1/6)^2 / (1/6)]
Where:
- n is the number of times the die is rolled (15 in this case)
- ∆p_j represents the difference between the observed frequency of each face j and the expected frequency 1/6 for a fair die.
Now, the number of degrees of freedom, k, is equal to the number of categories being tested minus 1. In this case, there are 6 categories (the 6 sides of the die), so k = 6 - 1 = 5.
Therefore, the value of k is equal to 5.