Question

complete parts (a) through (c) below
(a) determine the critical value(s) for a right - tailed test of a population mean at the \\( \alpha=0.10 \\) level of significance with 19 degrees of freedom.
(b) determine the critical value(s) for a left - tailed test of a population mean at the \\( \alpha = 0.01 \\) level of significance based on a sample size of \\( n = 20 \\).
(c) determine the critical value(s) for a two - tailed test of a population mean at the \\( \alpha=0.01 \\) level of significance based on a sample size of \\( n = 14 \\).
click here to view the t - distribution area in right tail
(a) \\( t_{\alpha}= \\) (round to three decimal places as needed.)
(b) \\( t_{\alpha}= \\) (round to three decimal places as needed.)
(c) \\( t_{\frac{\alpha}{2}}= \\) (round to three decimal places as needed.)

Explanation:

Step1: Determine degrees of freedom for each part

• For part (a): Degrees of freedom \(df=n - 1=19-1 = 18\)
• For part (b): Degrees of freedom \(df=n - 1=20 - 1=19\)
• For part (c): Degrees of freedom \(df=n - 1=14 - 1=13\)

Step2: Use t - distribution table for critical values

• Part (a): Right - tailed test with \(\alpha=0.10\) and \(df = 18\). Looking at the t - distribution table (area in right - tail), the critical value \(t_{\alpha}\) is such that \(P(T>t_{\alpha})=\alpha\). From the table, \(t_{0.10,18}=1.330\)
• Part (b): Left - tailed test with \(\alpha = 0.01\) and \(df=19\). For a left - tailed test, the critical value \(t_{\alpha}\) satisfies \(P(T<t_{\alpha})=\alpha\). Since the t - distribution is symmetric, \(t_{0.01,19}=- 2.539\)
• Part (c): Two - tailed test with \(\alpha=0.01\), so \(\frac{\alpha}{2}=0.005\) and \(df = 13\). The critical values \(t_{\frac{\alpha}{2}}\) satisfy \(P(T < - t_{\frac{\alpha}{2}})+P(T>t_{\frac{\alpha}{2}})=\alpha\). From the table, \(t_{0.005,13}=\pm3.012\)

Answer:

(a) \(1.330\)
(b) \(-2.539\)
(c) \(\pm3.012\)