Final answer:
A result is considered significant in statistics if the p-value is less than the predetermined alpha level, indicating strong evidence against the null hypothesis. In the provided situations, a significant result is reflected when the p-value < alpha, and all scenarios where the decision is to reject the null hypothesis indicate significant results.
Step-by-step explanation:
In statistics, a result is considered significant if the evidence is strong enough to support a hypothesis. This is often determined by looking at the p-value which is a measure of how extreme the observed results would be if the null hypothesis were true. You know you have a significant result if:
- The p-value is less than the pre-determined alpha level (α), signifying that the observed data are unlikely under the null hypothesis.
- t-statistic is beyond a certain critical value that corresponds to the chosen significance level, although this wasn't directly an option provided by your question.
- The confidence interval does not contain the null value (usually 0 for a mean difference or 1 for a ratio), and is generally smaller, not larger, for significant results.
Based on your provided scenarios:
- Reflective of a significant result is when the p-value < alpha (b).
- Not reflective of a significant result is when we 'Do not reject the null hypothesis' because the p-value or t-statistic does not exceed the threshold set by alpha.
In all of the scenarios given, when the decision is to 'Reject the null hypothesis,' it indicates that the results were statistically significant, meaning there was enough evidence to suggest an effect or difference that was not due to random chance at the specified alpha level.