16. The table shows the number of votes by fans for contestants on a popular television show. The observed counts are: Contestant A: 120, B: 136, C: 144. Which value represents the degrees of freedom if a goodness-of-fit test is performed?
Answer: B
The degrees of freedom for a goodness-of-fit test is 2.
In a goodness-of-fit test, the degrees of freedom can be calculated using the formula: \( df = k - 1 \), where \( k \) is the number of categories. In this case, there are three contestants (A, B, and C), so the degrees of freedom is \( 3 - 1 = 2 \).
A) 1
Option A is incorrect because a degrees of freedom of 1 would imply that there are only two categories being compared. Since there are three contestants in this scenario, this option does not apply.
B) 2
Option B is correct as it accurately reflects the calculation of degrees of freedom for a goodness-of-fit test with three categories. The formula \( df = k - 1 \) leads to \( 3 - 1 = 2 \), confirming this choice as valid.
C) 5
Option C is incorrect because a degrees of freedom value of 5 would suggest that there are six categories under consideration. Given only three contestants are present, this option exceeds the correct calculation.
D) 6
Option D is also incorrect. A degrees of freedom of 6 implies that there are seven categories, which is not the case here, as there are only three contestants in the analysis.
Conclusion
The correct answer is option B, which represents the appropriate degrees of freedom for a goodness-of-fit test involving three contestants. All other options fail to accurately reflect the calculation based on the number of categories involved in the test. This illustrates the importance of understanding the relationship between categories and degrees of freedom in statistical analysis.