Step 1 :The critical value is the point (or points) on the scale of the test statistic beyond which we reject the null hypothesis, and, is derived from the level of significance α of the test.
Step 2 :For a right-tailed test, the critical value is the value such that the area to its right under the curve is equal to α. For a left-tailed test, the critical value is the value such that the area to its left under the curve is equal to α. For a two-tailed test, the critical values are the values such that the area to their left and right under the curve is equal to α/2.
Step 3 :For part (a), the critical value for a right-tailed test at the α=0.10 level of significance with 10 degrees of freedom is \(t_{\text {crit }}=+1.372\).
Step 4 :For part (b), we need to find the critical value for a left-tailed test with α=0.01 and n=15. The degrees of freedom is n-1=14. The critical value is \(t_{\text {crit }}=\boxed{-2.624}\) (rounded to three decimal places).
Step 5 :For part (c), we need to find the critical values for a two-tailed test with α=0.10 and n=11. The degrees of freedom is n-1=10. The critical values are \(t_{\text {crit }}=\boxed{-1.812}\) and \(\boxed{1.812}\) (rounded to three decimal places).