To find the critical value [tex]\( z_{\alpha/2} \)[/tex] for a given confidence level, we need to follow these steps:
1. Determine the value of [tex]\( \alpha \)[/tex]:
The confidence level is 88%, which means that 88% of the data falls within the confidence interval. Therefore:
[tex]\[
\text{confidence level} = 0.88
\][/tex]
The value of [tex]\( \alpha \)[/tex] is:
[tex]\[
\alpha = 1 - \text{confidence level} = 1 - 0.88 = 0.12
\][/tex]
2. Divide [tex]\( \alpha \)[/tex] by 2:
Since the confidence interval is two-tailed (symmetric around the mean), we need to split [tex]\( \alpha \)[/tex] into two tails. Thus:
[tex]\[
\alpha / 2 = 0.12 / 2 = 0.06
\][/tex]
3. Find the critical value [tex]\( z_{\alpha/2} \)[/tex]:
The critical value [tex]\( z_{\alpha/2} \)[/tex] corresponds to the value above which [tex]\( \alpha / 2 \)[/tex] (or 6%) of the data lies in the standard normal distribution (Z-distribution). This value can be found using statistical tables or a computational tool.
In this case, the critical value corresponding to [tex]\( \alpha / 2 = 0.06 \)[/tex] is approximately:
[tex]\[
z_{0.06} \approx 1.5547735945968535
\][/tex]
4. Round to two decimal places:
Finally, rounding the critical value to two decimal places, we get:
[tex]\[
z_{\alpha/2} \approx 1.55
\][/tex]
Therefore, for a confidence level of 88%, the critical value [tex]\( z_{\alpha/2} \)[/tex] is:
[tex]\[
z_{\alpha/2} = 1.55
\][/tex]
