To find the critical value \( t_{\alpha/2} \) for a confidence level of 90% with a sample size of 23, you'll first need to determine the degrees of freedom.
The degrees of freedom (df) is calculated as:
\[ \text{df} = n - 1 \]
where \( n \) is the sample size. For a sample size of 23:
\[ \text{df} = 23 - 1 = 22 \]
Next, for a 90% confidence level, the significance level \( \alpha \) is:
\[ \alpha = 1 - 0.90 = 0.10 \]
Since we need \( t_{\alpha/2} \), we divide \( \alpha \) by 2:
\[ \alpha/2 = 0.10 / 2 = 0.05 \]
Now, we will look up the critical value in the t-distribution table for 22 degrees of freedom and \( \alpha/2 = 0.05 \).
From the t-distribution table, the critical value \( t_{0.05} \) for 22 degrees of freedom is approximately:
\[ t_{0.05} \approx 1.645 \]
Therefore, the value of \( t_{\alpha/2} \) is:
\[ t_{\alpha/2} = 1.645 \]
Rounded to three decimal places, it is:
\[ \boxed{1.645} \]