Answer :
To determine the level of confidence corresponding to a given Z-score value [tex]z_{\alpha/2}[/tex], you need to understand its role in statistics, particularly in the context of confidence intervals.
Understanding [tex]z_{\alpha/2}[/tex]
Z-Score Explanation: The [tex]z_{\alpha/2}[/tex] value is the critical value used in a standard normal distribution. It represents the cutoff point for the alpha level divided by 2 (i.e., [tex]\alpha/2[/tex]), which leaves equal tails in both ends of the distribution. This corresponds to the probability in the tails beyond this [tex]z[/tex]-score.
Finding the Confidence Level: To find the confidence level, you use the standard normal distribution table or use a calculator that provides the probability that lies to the left of the [tex]z[/tex]-score.
Z-Score of 1.64: For [tex]z_{\alpha/2} = 1.64[/tex], the area to the left is approximately 0.9495.
Calculating the Confidence Level:
- Subtract the left tail probability from 1: [tex]1 - 0.9495 = 0.0505[/tex].
- This implies that the two tails combined have a probability of 0.0505, split equally across both tails.
- Thus, calculate the confidence level: [tex]1 - (2 \times 0.02525) = 0.90[/tex].
Hence, a [tex]z_{\alpha/2}[/tex] value of 1.64 corresponds to a 90% confidence level. This means that if you construct a confidence interval using this [tex]z[/tex]-score, it would capture the true parameter 90% of the time if the same sampling method were repeated.
