For $$ C=95 \% $$ then $$ Z_{95}=2.56 $$

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1. We begin with a confidence level of $$ C = 95\% $$. For many applications (when constructing a two‐tailed confidence interval) the confidence level indicates that
   $$
   \alpha = 1 - 0.95 = 0.05,
   $$
   and each tail has an area of
   $$
   \frac{\alpha}{2} = 0.025.
   $$

2. The corresponding critical value is found by solving
   $$
   P(Z \le z_{1-\alpha/2}) = 0.975.
   $$
   Looking up the standard normal distribution or using a calculator gives
   $$
   z_{1-\alpha/2} \approx 1.96.
   $$

3. The statement $$ Z_{95} = 2.56 $$ does not match the usual critical value for a 95% two‐tailed confidence interval. One should note:
   - For a two‐tailed 95% confidence interval, the correct critical value is approximately $$ 1.96 $$.
   - If one interprets “$$Z_{95}$$” as the quantile for which $$ P(Z \le z) = 0.95 $$ (a one‐tailed situation), then the value would be about
     $$
     z \approx 1.645.
     $$
   - The value $$ 2.56 $$ is much closer to the critical value $$ \approx 2.575 $$ used for a 99% two‐tailed confidence interval.

4. In summary, for a 95% confidence level (two‐tailed):
   $$
   z_{0.975} \approx 1.96,
   $$
   so the statement $$ Z_{95} = 2.56 $$ is not the standard value used in this context.
For a 95% confidence level, the standard critical value is approximately 1.96, not 2.56.

For \( C=95 \% \) then \( Z_{95}=2.56 \)

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