To find the critical value $z_{\frac{\alpha}{2}}$ for a confidence level of 91%, we can follow these steps:

1. The confidence level is given as 91%, which means the remaining area under the normal distribution (the significance level $\alpha$) is: $\alpha = 1 - 0.91 = 0.09$

2. Since we are looking for the critical value $z_{\frac{\alpha}{2}}$, we need to divide $\alpha$ by 2: $\frac{\alpha}{2} = \frac{0.09}{2} = 0.045$

3. Now, we need to find the $z$-score that corresponds to the cumulative probability of $1 - 0.045 = 0.955$ (since we are looking for the value on the right side of the standard normal distribution).

4. Using a standard normal distribution table or a calculator, the $z$-score that corresponds to a cumulative probability of 0.955 is approximately: $z_{\frac{\alpha}{2}} \approx 1.70$

Therefore, the critical value $z_{\frac{\alpha}{2}}$ for a 91% confidence level is approximately 1.70.

Would you like more details or have any other questions?

Here are 5 related questions to expand on this:

1. How is the critical value used in hypothesis testing?
2. What is the significance level $\alpha$, and how does it relate to confidence levels?
3. How do you calculate confidence intervals using the critical value?
4. What is the difference between a one-tailed and a two-tailed test in hypothesis testing?
5. Can you explain the relationship between the z-score and the area under the normal distribution curve?

Tip: Always double-check your confidence level and significance level to ensure you are using the right side of the normal distribution.