A fair 20-sided die is rolled 60 times, and the value of chi-square is computed using expected counts of 3 for each face. If this process is repeated many times, the shape of the distribution of the values of chi-square should be...
A uniform
B bimodal
C skewed left
D skewed right
E approximately normal
To determine the shape of the distribution of the values of chi-square in this scenario, we need to understand the concept of chi-square and its expected behavior.
The chi-square test is a statistical test used to determine if there is a significant association between two categorical variables. It calculates the difference between observed and expected frequencies and estimates the probability that the observed frequencies differ from the expected frequencies by chance.
In this case, a fair 20-sided die is rolled 60 times, and the chi-square value is computed using expected counts of 3 for each face. The expected counts of 3 for each face means that if the die were rolled 60 times, we would expect each face to come up approximately 3 times.
When this process is repeated many times, the shape of the distribution of the values of chi-square depends on the number of degrees of freedom. In this case, since we are considering a 20-sided die with 20 possible outcomes, there would be 20-1 = 19 degrees of freedom.
The shape of the distribution of the values of chi-square with a specific number of degrees of freedom is approximately normal when the sample size is large. As the number of degrees of freedom increases, the chi-square distribution becomes more symmetric and bell-shaped.
Therefore, the correct answer is E) approximately normal.