To determine the valid conclusion based on the given [tex]$z$[/tex]-statistic and the critical [tex]$z$[/tex]-values, we need to compare the calculated [tex]$z$[/tex]-statistic to the range defined by the critical [tex]$z$[/tex]-values. Here’s a detailed step-by-step explanation:
1. Understand the Hypotheses:
- The null hypothesis (H₀) is typically the hypothesis that there is no effect or no difference. It is what you seek to reject when performing a hypothesis test.
- The alternate hypothesis (H₁) is what you are looking to provide evidence for. It is usually that there is an effect or a difference.
2. Identify the [tex]$z$[/tex]-statistic:
- Tyesha’s calculated [tex]$z$[/tex]-statistic is [tex]\(2.1\)[/tex].
3. Identify the critical [tex]$z$[/tex]-values:
- The critical [tex]$z$[/tex]-values provided are [tex]\(-1.96\)[/tex] and [tex]\(1.96\)[/tex].
4. Define the Decision Rule:
- In hypothesis testing, we compare the [tex]$z$[/tex]-statistic to the critical [tex]$z$[/tex]-values to make a decision:
- If the [tex]$z$[/tex]-statistic is within the range defined by the critical values [tex]\([-1.96, 1.96]\)[/tex], we fail to reject the null hypothesis.
- If the [tex]$z$[/tex]-statistic is outside the range defined by the critical values, we reject the null hypothesis.
5. Compare the [tex]$z$[/tex]-statistic to the critical values:
- We need to check if [tex]\(2.1\)[/tex] falls within or outside the range [tex]\([-1.96, 1.96]\)[/tex].
6. Make the Decision:
- Since [tex]\(2.1\)[/tex] is greater than [tex]\(1.96\)[/tex] and does not lie within the range [tex]\([-1.96, 1.96]\)[/tex], we have:
- [tex]\(2.1 > 1.96\)[/tex] (outside the range)
7. Draw the Conclusion:
- Because the [tex]$z$[/tex]-statistic ([tex]\(2.1\)[/tex]) is outside the critical value range ([tex]\([-1.96, 1.96]\)[/tex]), we have sufficient evidence to reject the null hypothesis.
Therefore, the valid conclusion based on these results is:
One can reject the null hypothesis.
