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(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.

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