Critical Z values are the number of standard deviations you need to go in either direction from your sample results to build a confidence interval. Your sample results are only an estimate of the true parameter p and thus you need to add a "fudge factor" to each side of this number. The technical term for this fudge factor is "margin of error" and is the product of your critical z (z*) multiplied by your standard error (s.e.). See the Canvas Lesson 6.1 and 6.2 on how to calculate a critical value.Find the critical values for the following levels of confidence.Level of ConfidenceCritical Z (Z*)Feedback95%90%99%86%70%What relationship do you see between your desired level of confidence and your critical z?

To find the critical Z values for different levels of confidence, we need to refer to the standard normal distribution table or use statistical software. The critical Z value (Z*) corresponds to the number of standard deviations from the mean required to capture the desired level of confidence.

Here are the steps to find the critical Z values for the given levels of confidence:

1. 95% Confidence Level:

   • For a 95% confidence level, the critical Z value is typically found by looking up the value that corresponds to the middle 95% of the standard normal distribution.
   • This leaves 2.5% in each tail (since 100% - 95% = 5%, and 5%/2 = 2.5%).
   • The critical Z value for 95% confidence is approximately 1.96.

2. 90% Confidence Level:

   • For a 90% confidence level, the critical Z value corresponds to the middle 90% of the standard normal distribution.
   • This leaves 5% in each tail (since 100% - 90% = 10%, and 10%/2 = 5%).
   • The critical Z value for 90% confidence is approximately 1.645.

3. 99% Confidence Level:

   • For a 99% confidence level, the critical Z value corresponds to the middle 99% of the standard normal distribution.
   • This leaves 0.5% in each tail (since 100% - 99% = 1%, and 1%/2 = 0.5%).
   • The critical Z value for 99% confidence is approximately 2.576.

4. 86% Confidence Level:

   • For an 86% confidence level, the critical Z value corresponds to the middle 86% of the standard normal distribution.
   • This leaves 7% in each tail (since 100% - 86% = 14%, and 14%/2 = 7%).
   • The critical Z value for 86% confidence is approximately 1.48.

5. 70% Confidence Level:

   • For a 70% confidence level, the critical Z value corresponds to the middle 70% of the standard normal distribution.
   • This leaves 15% in each tail (since 100% - 70% = 30%, and 30%/2 = 15%).
   • The critical Z value for 70% confidence is approximately 1.04.

Relationship between Desired Level of Confidence and Critical Z: The relationship observed is that as the desired level of confidence increases, the critical Z value also increases. This is because a higher confidence level means that we want to capture a larger portion of the distribution, which requires extending further from the mean in both directions. Conversely, a lower confidence level requires a smaller critical Z value, as we are capturing a smaller portion of the distribution.