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Chapter 9: Problem 9

Determine the critical value \(z_{\alpha / 2}\) that corresponds to the givenlevel of confidence. \(98 \%\)

### Short Answer

Expert verified

The critical value \(z_{\alpha / 2}\) for a 98% confidence level is approximately 2.33.

## Step by step solution

01

## Identify the given confidence level

The given level of confidence is 98%. This means we are 98% confident that our interval estimate will contain the true population parameter.

02

## Convert confidence level to significance level

The significance level \(\alpha\) is calculated as \(\alpha = 1 - \text{confidence level}\). So, \alpha = 1 - 0.98 = 0.02\.

03

## Divide the significance level by 2

Since the critical value \(z_{\alpha / 2}\) corresponds to the area in each tail of the standard normal distribution curve, divide the significance level by 2: \alpha / 2 = 0.02 / 2 = 0.01\.

04

## Find the critical value

Using the standard normal distribution table or a calculator, find the critical value that corresponds to an area of 0.01 in the tail. The critical value \(z_{\alpha / 2}\) for \alpha / 2 = 0.01\ is approximately 2.33.

## Key Concepts

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

###### confidence level

The confidence level is a crucial concept in statistics that shows how confident we are in a statistical estimate. In other words, it tells us how often we can expect our interval estimate to contain the true population parameter if we were to take many samples.

The confidence level is typically represented as a percentage. Common confidence levels include 90%, 95%, and 98%.

- A 98% confidence level means we can be 98% sure our interval will include the true population parameter. There is a 2% chance it will not.
- The higher the confidence level, the more confident we are in our estimate. However, higher confidence levels usually result in wider intervals.

Understanding confidence levels helps us decide how precise we want our estimates to be.

###### significance level

In statistics, the significance level \alpha\ (alpha) represents the probability of rejecting the null hypothesis when it is actually true. It is defined as \( \alpha = 1 - \text{confidence level} \) .

In the given exercise, the confidence level is 98%, so:

\( \alpha = 1 - 0.98 = 0.02 \)

This means there is a 2% chance of making a mistake in rejecting a true null hypothesis.

- The significance level indicates how stringent we are with our hypothesis testing. A lower alpha means more stringent criteria.
- For instance, a 0.05 significance level is commonly used, leading to a 5% risk of concluding that an effect exists when it does not.

Knowing the significance level helps us understand the risk involved in our statistical hypothesis tests.

###### standard normal distribution

The standard normal distribution is a special type of normal distribution with a mean of 0 and a standard deviation of 1. It is crucial for statistical calculations because it provides a common framework.

The standard normal distribution is symmetrical and has a bell shape. The total area under the curve is 1, representing the total probability.

- The z-values in this distribution tell us how many standard deviations a point is from the mean.
- Critical values in hypothesis testing are often determined using the standard normal distribution.

In the exercise, we use the standard normal distribution table to find the critical value that corresponds to a specific significance level.

###### critical value calculation

Calculating the critical value is an essential step in hypothesis testing that defines the boundary for deciding whether to reject the null hypothesis.

Here's how we determine the critical value in the given exercise:

- The confidence level is 98%, meaning \( \alpha = 0.02 \).
- We then divide the significance level by 2 to get \alpha / 2 = 0.01\.
- Next, we use a standard normal distribution table or a calculator to find the z-value for \alpha / 2 = 0.01 \.
- The critical value, \( z_{\alpha / 2} = 2.33 \), tells us where our data would fall to be considered unusual compared to a normal population.

This critical value helps us make decisions in statistical tests, indicating the point beyond which we reject the null hypothesis.

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