Determine the critical values for these tests of a population standard deviation. (a) A right-tailed test with 12 degrees of freedom at the $\alpha=0.05$ level of significance (b) A left-tailed test for a sample of size $n=26$ at the $\alpha=0.01$ level of significance (c) A two-tailed test for a sample of size $n=26$ at the $\alpha=0.1$ level of significance Click the icon to view a table/a critical values for the Chi-Square Distribution. (a) The critical value for this right-tailed test is 21.026 . (Round to three decimal places as needed.) (b) The critical value for this left-tailed test is (Round to three decimal places as needed.)
Solution
Step 1 ：Given a right-tailed test with 12 degrees of freedom at the \(\alpha=0.05\) level of significance, the critical value is the value such that the area to the right of it under the chi-square distribution curve is equal to the significance level.
Step 2 ：Using a chi-square distribution table or a statistical function, the critical value for this test is found to be approximately 21.026.
Step 3 ：Given a left-tailed test with a sample size of 26 (which means 25 degrees of freedom) at the \(\alpha=0.01\) level of significance, the critical value is the value such that the area to the left of it under the chi-square distribution curve is equal to the significance level.
Step 4 ：Using a chi-square distribution table or a statistical function, the critical value for this test is found to be approximately 11.524.
Step 5 ：Given a two-tailed test with a sample size of 26 (which means 25 degrees of freedom) at the \(\alpha=0.1\) level of significance, the critical values are the values such that the area to the left of the lower value and to the right of the higher value under the chi-square distribution curve is equal to the significance level divided by 2.
Step 6 ：Using a chi-square distribution table or a statistical function, the critical values for this test are found to be approximately 14.611 and 37.652.
Step 7 ：Final Answer: \(\boxed{21.026}\) is the critical value for the right-tailed test, \(\boxed{11.524}\) is the critical value for the left-tailed test, and \(\boxed{14.611}\) and \(\boxed{37.652}\) are the critical values for the two-tailed test.
