Finding Critical Values

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

99.5%

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Step 1: Understand the problem. The critical value z_{α/2} is a z-score that corresponds to the given confidence level. For a 99.5% confidence level, the area in the middle of the standard normal distribution is 0.995, and the remaining area (α) is split equally between the two tails.

Step 2: Calculate α. Subtract the confidence level from 1 to find the total area in the tails: α = 1 - 0.995 = 0.005.

Step 3: Divide α by 2 to find the area in one tail: α/2 = 0.005 / 2 = 0.0025. This represents the area in the left tail of the standard normal distribution.

Step 4: Use a z-table or statistical software to find the z-score that corresponds to the cumulative area of 1 - α/2 = 1 - 0.0025 = 0.9975. This cumulative area represents the area to the left of the critical value z_{α/2}.

Step 5: Interpret the result. The z-score you find is the critical value z_{α/2} for the 99.5% confidence level. This value is symmetric, so the critical values are ±z_{α/2}.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 many standard deviations away from the mean a data point must be to fall within a certain confidence level.

Confidence Level

The confidence level represents the probability that the confidence interval will contain the true population parameter. It is expressed as a percentage, such as 99.5%, indicating that if we were to take many samples and build a confidence interval from each, approximately 99.5% of those intervals would contain the true parameter. Higher confidence levels result in wider intervals.

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. In the context of finding critical values, the z-score corresponding to a specific confidence level helps determine the cutoff points for the tails of the normal distribution, which are essential for constructing confidence intervals.

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