For conducting a two-tailed hypothesis test with a certain data set, using the smaller of $n_{1}-1$ and $n_{2}-1$ for the degrees of freedom results in $\mathrm{df}=11$, and the corresponding critical values are $t= \pm 2.201$. Using the formula for the exact degrees of freedom results in $\mathrm{df}=19.063$, and the corresponding critical values are $t= \pm 2.093$. How is using the critical values of $t= \pm 2.201$ more "conservative" than using the critical values of \pm 2.093 ? Choose the correct answer below. A. Using the critical values of $t= \pm 2.201$ results in rounding the test statistic to more decimal places than using the critical values of \pm 2.093 . B. Using the critical values of $t= \pm 2.201$ requires fewer calculations than using the critical values of \pm 2.093 . C. Using the critical values of $t= \pm 2.201$ is more likely to lead to rejection of the null hypothesis than using the critical values of \pm 2.093 . D. Using the critical values of $t= \pm 2.201$ is less likely to lead to rejection of the null hypothesis than using the critical values of \pm 2.093 .
Solution
Step 1 ：In a two-tailed hypothesis test, the critical values are the points beyond which we reject the null hypothesis. The larger the absolute value of the critical value, the further it is from the mean, and the less likely we are to reject the null hypothesis. Therefore, using a larger critical value is more 'conservative' because it reduces the likelihood of rejecting the null hypothesis when it is true (Type I error).
Step 2 ：In this case, the critical value of \(t= \pm 2.201\) is larger than \(t= \pm 2.093\). Therefore, using the critical values of \(t= \pm 2.201\) is more 'conservative' because it is less likely to lead to rejection of the null hypothesis than using the critical values of \(t= \pm 2.093\).
Step 3 ：\(\boxed{\text{The correct answer is D. Using the critical values of } t= \pm 2.201 \text{ is less likely to lead to rejection of the null hypothesis than using the critical values of } t= \pm 2.093}\)
