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?

Rationale

Degrees of freedom for a goodness-of-fit test is calculated as the number of categories minus one.

In this case, there are four brands (A, B, C, D), which means we have four categories. Therefore, the degrees of freedom for the goodness-of-fit test is calculated as 4 - 1 = 3.

A (1) YOUR ANSWER

This option suggests that there is only one degree of freedom, which would apply if there were two categories in the test. However, with four brands involved, the calculation of degrees of freedom must account for the multiple categories, leading to a higher count than one.

B (3) CORRECT

This is the correct answer. The degrees of freedom for a goodness-of-fit test are determined by subtracting one from the number of categories, which in this case is 4 (the brands). Therefore, 4 - 1 equals 3, reflecting the variance in the observed data across the different brands.

C (7) YOUR ANSWER

This choice incorrectly implies that there are more degrees of freedom than there actually are. A value of 7 degrees of freedom would suggest there are 8 categories to analyze, which does not align with the four brands presented in the survey.

D (8) YOUR ANSWER

This option suggests eight degrees of freedom, which would require nine categories. Since there are only four brands represented in the survey, this value is not applicable and indicates a misunderstanding of how to calculate degrees of freedom in this context.

Conclusion

In a goodness-of-fit test, the degrees of freedom are calculated as the number of categories minus one. Given that there are four brands in the survey data, the appropriate calculation yields three degrees of freedom. Understanding this concept is crucial for accurately interpreting statistical test results and ensuring sound conclusions from data analysis.

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?

Degrees of freedom for a goodness-of-fit test is calculated as the number of categories minus one.

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