Finding Critical Values.


In Exercises 5–8, find the critical value z=a/2 that corresponds to the given confidence level.


90%

Verified step by step guidance

1. Step 1: Understand the problem. The critical value zₐ/₂ corresponds to the z-score that separates the middle area (confidence level) from the tails in a standard normal distribution. For a 90% confidence level, the middle area under the curve is 0.90, leaving 0.10 in the tails.
2. Step 2: Divide the remaining area (0.10) equally between the two tails. Since the standard normal distribution is symmetric, each tail will have an area of 0.05.
3. Step 3: Determine the cumulative area to the left of the critical value zₐ/₂. For the upper critical value, this area is 1 - 0.05 = 0.95. For the lower critical value, the cumulative area is 0.05.
4. Step 4: Use a z-table or statistical software to find the z-score corresponding to the cumulative area of 0.95 (upper critical value) and 0.05 (lower critical value). These z-scores are the critical values.
5. Step 5: Interpret the result. The critical values zₐ/₂ are symmetric around the mean (0) of the standard normal distribution. The positive z-score corresponds to the upper critical value, and the negative z-score corresponds to the lower critical value.

Key Concepts

Critical Value

A critical value is a point on the scale of the test statistic that separates the region where the null hypothesis is rejected from the region where it is not rejected. In the context of confidence intervals, it corresponds to the z-score that captures the desired level of confidence, indicating how far from the mean we need to go to encompass a certain percentage of the data.

Confidence Level

The confidence level represents the probability that the confidence interval will contain the true population parameter. Common confidence levels are 90%, 95%, and 99%. A 90% confidence level means that if we were to take many samples and build a confidence interval from each, approximately 90% of those intervals would contain the true parameter.

Z-Score

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values, expressed in terms of standard deviations. In the context of finding critical values, z-scores are used to determine the cutoff points for the desired confidence level, allowing statisticians to understand how extreme a value is within a normal distribution.