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Find the critical value zva/2 needed to construct a confidence interval with level 99.5%

3 answers

To find the critical value \( z_{\alpha/2} \) for a confidence level of 99.5%, we need to determine the value that corresponds to the tails of the standard normal distribution.

  1. Identify the significance level:

    • A confidence level of 99.5% has a significance level \( \alpha = 1 - 0.995 = 0.005 \).
    • Since \( z_{\alpha/2} \) represents the critical value for the upper tail of the standard normal distribution, we need to find \( \alpha/2 \): \[ \alpha/2 = 0.005 / 2 = 0.0025 \]

  2. Find the critical z-value:

    • We are looking for the z-value such that the area to the right of it is 0.0025. This is equivalent to finding the z-value that leaves 0.9975 to the left: \[ P(Z < z_{\alpha/2}) = 0.9975 \]

  3. Using the standard normal distribution table or calculator:

    • Using a standard normal distribution table or an inverse normal calculator, we look for the z-value that corresponds to 0.9975.
    • This value is typically found to be approximately \( z_{0.0025} \approx 2.576 \).

Therefore, the critical value \( z_{\alpha/2} \) needed to construct a confidence interval with a confidence level of 99.5% is: \[ z_{\alpha/2} \approx 2.576 \]

how do you use a TI-84 calculator to find this

To find the critical value \( z_{\alpha/2} \) for a confidence level of 99.5% using a TI-84 calculator, you can follow these steps:

  1. Turn on your TI-84 calculator.

  2. Access the distribution menu:

    • Press the 2nd button, then VARS (which is the DISTR option). This will open the distribution menu.

  3. Select the inverse normal function:

    • Scroll down to invNorm( (often option 2). Select it by pressing ENTER.

  4. Input the area:

    • Since you need to find \( z_{\alpha/2} \) for a right tail area of \( 0.0025 \), you need to input the cumulative area to the left of the critical value. As explained earlier, this is \( 1 - 0.0025 = 0.9975 \).
    • Type in 0.9975 as the argument: \[ invNorm(0.9975) \]

  5. Press ENTER:

    • After typing in invNorm(0.9975), press ENTER to get the result.

  6. Interpret the result:

    • The calculator will display the \( z \)-value corresponding to the area of 0.9975, which will be roughly \( 2.576 \).

Example Input on the TI-84:

  1. 2nd -> VARS
  2. Scroll to 2:invNorm( and press ENTER
  3. Type 0.9975 and then press ), so it reads invNorm(0.9975)
  4. Press ENTER to see the output.

This should yield the critical z-value you need for a 99.5% confidence level.