High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ (c) What value of [tex]z_{\alpha/2}[/tex] in the confidence interval (CI) formula below results in a confidence level of 99.3%?

[tex]x - z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}, \, x + z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}[/tex]

To find the value of [tex]z_{\alpha/2}[/tex] that corresponds to a confidence level of 99.3%, first determine the value of [tex]\alpha[/tex]. Recall that the confidence level is determined using the formula [tex]100(1 - \alpha)\%[/tex]. Solve this formula for [tex]\alpha[/tex].

[tex]100(1 - \alpha) = 99.3[/tex]

[tex]\alpha =[/tex]

The critical value [tex]z_{\alpha/2}[/tex] uses [tex]\alpha/2[/tex], which is [tex]\alpha/2 =[/tex]

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.

  1. The confidence level is given as 99.3%, which can be expressed in decimal form as 0.993.

  2. The formula relating confidence level and [tex]\alpha[/tex] is:
    [tex]100(1 - \alpha) = 99.3[/tex]

  3. 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]

  4. The critical value [tex]z_{\alpha/2}[/tex] involves [tex]\alpha/2[/tex]. Therefore:
    [tex]\alpha/2 = 0.007/2 = 0.0035[/tex]

  5. 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.