(b) For a confidence level of $90 \%$ and a sample size of 36 , find the critical value $t_{\alpha / 2}$. Round your answer to 3 places after the decimal point. \[ t_{\alpha / 2}= \]
Solution
Step 1 ：The critical value \(t_{\alpha / 2}\) is a value from the t-distribution that we use in hypothesis testing and confidence intervals. It depends on the desired level of confidence and the degrees of freedom, which is related to the sample size.
Step 2 ：In this case, the confidence level is \(90\%\) and the sample size is 36, so the degrees of freedom is \(36 - 1 = 35\).
Step 3 ：The value of \(t_{\alpha / 2}\) is found by looking up the appropriate value in a t-distribution table, or by using a statistical function.
Step 4 ：We want to find the value of \(t_{\alpha / 2}\) such that the area to the right of \(t_{\alpha / 2}\) (in the upper tail of the t-distribution) is \(\alpha / 2\), where \(\alpha = 1 - \text{confidence level} = 1 - 0.90 = 0.10\). So, \(\alpha / 2 = 0.10 / 2 = 0.05\).
Step 5 ：The critical value \(t_{\alpha / 2}\) for a \(90\%\) confidence level and a sample size of 36 is approximately \(1.69\). This is the value from the t-distribution that we would compare a test statistic to in order to make a decision in a hypothesis test, or that we would use to calculate a confidence interval.
Step 6 ：Final Answer: The critical value \(t_{\alpha / 2}\) for a \(90\%\) confidence level and a sample size of 36 is approximately \(\boxed{1.69}\).
