Answer :
To determine the value of [tex]z_{\alpha/2}[/tex] for a confidence level of 99.3%, we first need to solve for [tex]\alpha[/tex] in the context of confidence intervals.
The confidence level is given as 99.3%, which can be expressed in decimal form as 0.993.
The formula relating confidence level and [tex]\alpha[/tex] is:
[tex]100(1 - \alpha) = 99.3[/tex]Solving for [tex]\alpha[/tex], we first convert the percentage to decimal form:
[tex]1 - \alpha = 0.993[/tex]
[tex]\alpha = 1 - 0.993 = 0.007[/tex]The critical value [tex]z_{\alpha/2}[/tex] involves [tex]\alpha/2[/tex]. Therefore:
[tex]\alpha/2 = 0.007/2 = 0.0035[/tex]To find [tex]z_{\alpha/2}[/tex], we look up the value in the standard normal distribution table or use a calculator that provides z-scores:
- For [tex]\alpha/2 = 0.0035[/tex], the corresponding z-score is approximately 2.75. This z-score represents the point beyond which 0.35% of the data lies in each tail of the normal distribution.
Final Result: For a 99.3% confidence level, the critical value [tex]z_{\alpha/2}[/tex] is approximately 2.75.