Final answer:
In a t-test, to determine if two means are significantly different, researchers compare the calculated t-value to the critical t-value. If the t-value is equal to or greater than the critical t-value at the chosen significance level (often 0.05), the null hypothesis is rejected, suggesting a significant difference between the means.
Step-by-step explanation:
Researchers use a t-test to determine if two means are significantly different from each other. The rule they apply is:
- Calculate the t-value (test statistic) based on the sample data.
- Determine the t(critical)-value (critical t-value) from the t-distribution table at a certain significance level (α), typically 0.05 for a 5% significance level.
- Compare the calculated t-value with the critical t-value.
If the absolute value of the t-value is equal to or greater than the t(critical)-value, then we reject the null hypothesis, indicating that there is significant evidence to suggest the two means are different. Otherwise, if the t-value is smaller than the t(critical)-value, we fail to reject the null hypothesis.
For example, with a significance level of α = 0.05 and a t-distribution with 29 degrees of freedom, the critical t-value is 2.045. If the calculated t-value is less than 2.045, we fail to reject the null hypothesis. If the calculated t-value is greater than or equal to 2.045, we reject the null hypothesis.
Overall, the correct rule from the options you've provided would be that two means are considered significantly different if the t(critical)-value is equal to or smaller than the t-value, option 3.