To determine the correct decision for the hypothesis test, we need to compare the obtained $t$-statistic ($t = 2.015$) with the critical values of $t$ for the given degrees of freedom ($df$) and significance levels ($\alpha = 0.05$ and $\alpha = 0.01$).

Step-by-Step Process:

1. Degrees of Freedom ($df$): For two independent samples, the degrees of freedom are: $df = (n_1 - 1) + (n_2 - 1) = (10 - 1) + (10 - 1) = 18$

2. Critical Values for a Two-Tailed Test: Using a $t$-distribution table or calculator for $df = 18$:

   • For $\alpha = 0.05$ (two-tailed): The critical $t$-values are $\pm 2.101$.
   • For $\alpha = 0.01$ (two-tailed): The critical $t$-values are $\pm 2.878$.

3. Comparison with $t = 2.015$:

   • For $\alpha = 0.05$: The obtained $t = 2.015$ is less than the critical value ($2.101$), so we fail to reject the null hypothesis.
   • For $\alpha = 0.01$: The obtained $t = 2.015$ is also less than the critical value ($2.878$), so we fail to reject the null hypothesis.

Conclusion:

The correct decision is:

Fail to reject the null hypothesis with either $\alpha = 0.05$ or $\alpha = 0.01$.

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Related Questions:

1. What does it mean to reject the null hypothesis in hypothesis testing?
2. How is the $t$-statistic calculated in an independent-measures $t$-test?
3. What are the assumptions required for an independent-measures $t$-test?
4. How does increasing sample size affect the critical $t$-value and decision-making?
5. What is the difference between one-tailed and two-tailed hypothesis tests?

Tip:

Always check the degrees of freedom and ensure you're using the correct $t$-distribution table for two-tailed or one-tailed tests.