54. 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: 129, B: 129, C: 142. A goodness-of-fit test was performed: ?� = 0.84, df = 2, and p-value = 0.6554. What is the appropriate conclusion?
Answer: C
Fail to reject the null hypothesis. There is not a significant difference among the brand preferences.
Based on the results of the goodness-of-fit test, the p-value of 0.6554 indicates that there is no statistically significant difference among the brand preferences. Therefore, we fail to reject the null hypothesis.
A) Reject the null hypothesis. There is a significant difference among the brand preferences.
This option is incorrect because rejecting the null hypothesis would imply that there is a statistically significant difference among the brand preferences, which is not supported by the p-value of 0.6554.
B) Reject the null hypothesis. There is not a significant difference among the brand preferences.
This option contradicts the statistical principles of hypothesis testing. If we reject the null hypothesis, it must indicate a significant difference, making this statement incorrect.
C) Fail to reject the null hypothesis. There is not a significant difference among the brand preferences.
This option correctly interprets the results of the goodness-of-fit test. Given the p-value of 0.6554, we fail to reject the null hypothesis, concluding there is no significant difference in brand preferences.
D) Fail to reject the null hypothesis. There is a significant difference among the brand preferences.
This option is incorrect because it incorrectly states that there is a significant difference despite the conclusion of the goodness-of-fit test indicating otherwise.
Conclusion
The correct answer, C, accurately reflects the results of the goodness-of-fit test, which shows no significant difference in brand preferences due to the high p-value. All other options misinterpret the statistical findings of the test, either suggesting an incorrect rejection of the null hypothesis or mislabeling the significance of the results.