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?
Fail to reject the null hypothesis. There is not a significant difference among the brand preferences.
The p-value of 0.6554 is greater than the common significance level of 0.05, indicating that the observed differences in brand preferences are not statistically significant. Therefore, we do not have enough evidence to reject the null hypothesis, which posits that there is no difference among the brand preferences.
This option is incorrect because the p-value of 0.6554 indicates a lack of significant evidence to reject the null hypothesis. A significant difference would require a p-value less than the chosen significance level, which is not the case here.
This choice contradicts itself; if we reject the null hypothesis, it implies that there is a significant difference. Therefore, this statement is logically inconsistent and cannot be correct.
This option is also incorrect because failing to reject the null hypothesis suggests that there is no significant difference among the brand preferences. Thus, stating that there is a significant difference while failing to reject it is contradictory.
In conclusion, the results of the goodness-of-fit test indicate that the observed preferences among the brands do not show statistically significant differences. Given the p-value of 0.6554, we appropriately fail to reject the null hypothesis and conclude that there is no significant difference in preferences among Brand A, Brand B, and Brand C.
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