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Chapter 7: Problem 6

Find the critical value \(z_{a / 2}\) that corresponds to the given confidence level. $$99 \%$$

Short Answer

Expert verified
The critical value {z_{a / 2}} for a 99% confidence level is approximately 2.575.

Step by step solution

01

Identify the Confidence Level

The first step is to note the confidence level given in the problem. Here, the confidence level is 99%. This will be used to determine the critical value.
02

Find Area in Each Tail

Since the confidence level is 99%, there is a 1% (or 0.01) total area in the tails of the standard normal distribution. Each tail will have half of this area. Therefore, the area in each tail is 0.01 / 2 = 0.005.
03

Locate the Critical Value

To find the critical value ({z_{a / 2}} ) for the area in each tail (0.005), use the standard normal (Z) table or appropriate statistical tools. The value found for 0.005 in one tail of the standard normal distribution corresponds to a Z-score that cuts off the top 0.5% of the distribution, which is approximately 2.575.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Level
The confidence level is a key concept in statistical inference. It represents the proportion of times that an estimated range of values would include the true population parameter if you repeated an experiment multiple times.

Here are some important points to understand about confidence levels:
  • Confidence level is usually expressed as a percentage. Common confidence levels include 90%, 95%, and 99%.
  • For a 99% confidence level, it means that if you were to take 100 different samples, approximately 99 of those samples would contain the true population mean.
  • Higher confidence levels yield broader confidence intervals, implying more uncertainty, while lower confidence levels yield narrower intervals.
This is important because it directly influences the critical value calculation that tells us how far from the mean our range goes to cover the desired proportion of the data.
Standard Normal Distribution
The standard normal distribution is a special case of the normal distribution. It has a mean of 0 and a standard deviation of 1.

Important aspects to remember about the standard normal distribution include:
  • It is symmetric about the mean. This means that the left side of the distribution is a mirror image of the right side.
  • The total area under the curve is 1, which represents the entire probability distribution.
  • In this distribution, data points are measured in terms of standard deviations from the mean. These measurements are known as Z-scores.
The standard normal distribution is used extensively in statistics because of its mathematical properties and its role in the calculation of Z-scores. By converting any normal distribution to a standard normal distribution, we can use Z-tables for finding probabilities and critical values efficiently.
Z-score
The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated as:
\[Z = \frac{X - \mu}{\sigma}\] where:
X is the value,
渭 is the mean,
蟽 is the standard deviation.

Understanding Z-scores is essential because:
  • Z-scores can tell us how far and in what direction a value deviates from the mean.
  • A Z-score tells you how many standard deviations a value is from the mean. A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it is below the mean.
  • Z-scores are crucial in determining critical values for hypothesis testing and confidence intervals.
In the context of the original exercise, the Z-score helps us determine the critical value, which in this example is approximately 2.575 for a 99% confidence level. This means that values beyond 2.575 standard deviations from the mean cover the top 0.5% of the distribution on either side.

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Most popular questions from this chapter

Find the critical value \(z_{a / 2}\) that corresponds to the given confidence level. $$90 \%$$

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