50. The results of a survey are shown in the table. The data shows the preferences for the same product offered by different brands. The observed counts are: Brand A: 122, B: 143, C: 119, D: 116. 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 3.
In a goodness-of-fit test, the degrees of freedom is calculated as the number of categories minus one. With four brands (A, B, C, and D), the degrees of freedom is 4 - 1, which equals 3.
A) 1
This option is incorrect because the degrees of freedom for a goodness-of-fit test requires subtracting one from the total number of categories. With four brands, using 1 would not accurately reflect the correct calculation.
B) 3
This option is correct since the degrees of freedom for a goodness-of-fit test is determined by taking the number of categories (4) and subtracting one, resulting in 3. This is the standard calculation used in hypothesis testing for categorical data.
C) 7
This option is incorrect as it suggests there are seven degrees of freedom, which would imply there are eight categories. However, there are only four brands in this case, making 7 an inaccurate representation of the degrees of freedom.
D) 8
This option is also incorrect because it implies that there are nine categories. The degrees of freedom calculation does not support this, as there are only four brands, leading to a maximum of three degrees of freedom.
Conclusion
The correct answer of 3 degrees of freedom is derived from the appropriate formula for calculating degrees of freedom in a goodness-of-fit test, which is the number of categories minus one. The other options fail to accurately reflect the calculation, highlighting a misunderstanding of how degrees of freedom are determined in this statistical context.