To solve this problem, let's carefully analyze the given statement from the research report and interpret its components step-by-step.
Given: A research report states: " [tex]\( t(22) = 1.71, p > 0.05 \)[/tex]."
1. Understanding the t-statistic and degrees of freedom (df):
- The report mentions [tex]\( t(22) = 1.71 \)[/tex].
- Here, [tex]\( t \)[/tex] is the t-statistic value, and [tex]\( 22 \)[/tex] is the degrees of freedom for the t-test.
2. Interpreting the p-value:
- The statement [tex]\( p > 0.05 \)[/tex] provides critical information about the p-value from the t-test.
- The p-value indicates the probability of obtaining test results at least as extreme as the one observed, assuming the null hypothesis is true.
3. P-value comparison with significance level:
- In hypothesis testing, we typically compare the p-value with a significance level [tex]\(\alpha\)[/tex] (commonly 0.05).
- If the p-value is less than or equal to [tex]\(\alpha\)[/tex], we reject the null hypothesis.
- If the p-value is greater than [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis.
Given [tex]\( p > 0.05 \)[/tex], it means that the p-value is greater than the typical significance level (0.05).
4. Conclusion based on the p-value:
- Since [tex]\( p > 0.05 \)[/tex], we do not have sufficient evidence to reject the null hypothesis.
- Therefore, the conclusion of the hypothesis test is to fail to reject the null hypothesis.
5. Determining the sample size (n):
- For a t-test, the degrees of freedom (df) can be calculated using the sample size (n) minus 1.
- Given that the degrees of freedom (df) are 22, we can deduce that the sample size is [tex]\( n = 22 + 1 = 23 \)[/tex].
Conclusion:
Based on the given research report, the correct conclusion is:
- To fail to reject the null hypothesis with a sample of [tex]\( n = 23 \)[/tex] participants.
So, the final answer is:
[tex]\[ \text{to fail to reject the null hypothesis with a sample of } n = 23 \text{ participants.} \][/tex]
