/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Find the critical value \(z_{a /... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the critical value \(z_{a / 2}\) that corresponds to the given confidence level. $$98 \%$$

Why does the bootstrap method require sampling with replacement? What would happen if we used the methods of this section but sampled without replacement?

Use the given data to find the minimum sample size required to estimate a population proportion or percentage. In a study of government financial aid for college students, it becomes necessary to estimate the percentage of full-time college students who earn a bachelor's degree in four years or less. Find the sample size needed to estimate that percentage. Use a 0.05 margin of error, and use a confidence level of \(95 \% .\) a. Assume that nothing is known about the percentage to be estimated. b. Assume that prior studies have shown that about \(40 \%\) of full-time students earn bachelor's degrees in four years or less. c. Does the added knowledge in part (b) have much of an effect on the sample size?

Use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion \(p ;(b)\) identify the value of the margin of error \(E ;(c)\) construct the confidence interval; (d) write a statement that correctly interprets the confidence interval. The drug Eliquis (apixaban) is used to help prevent blood clots in certain patients. In clinical trials, among 5924 patients treated with Eliquis, 153 developed the adverse reaction of nausea (based on data from Bristol-Myers Squibb Co.). Construct a \(99 \%\) confidence interval for the proportion of adverse reactions.

Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) Find the critical value \(t_{\alpha / 2}\) (b) find the critical value \(z_{\alpha / 2},\) or \((c)\) state that neither the normal distribution nor the \(t\) distribution applies. Here are summary statistics for randomly selected weights of newborn girls: \(n=205, \bar{x}=30.4 \mathrm{hg}, s=7.1 \mathrm{hg}\) (based on Data Set 4 "Births" in Appendix B). The confidence level is \(95 \%\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.