/*! 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 90 Understanding the \(z\) notation... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 90

Understanding the \(z\) notation, \(z(\alpha),\) requires us to know whether we have a \(z\) -score or an area. Each of the following expressions uses the \(z\) notation in a variety of ways, some typical and some not so typical. Find the value asked for in each of the following, and then with the aid of a diagram explain what your answer represents. a. \(\quad z(0.08)\) b. the area between \(z(0.98)\) and \(z(0.02)\) c. \(\quad z(1.00-0.01)\) d. \(\quad z(0.025)-z(0.975)\)

Short Answer

Expert verified
{a. \(-1.41\), b. \(0.96\), c. \(2.33\), d. \(3.92\)}. In terms of what these represent, they represent the z-scores (number of standard deviations from the mean) that correspond to the given probabilities in a standard normal distribution. For example, \(z(0.08)\) represents the z-score that is 0.08 probability to the left of the mean in a standard normal distribution.

Step by step solution

01

Find the Z-score for \(z(0.08)\)

This is asking for the z-score that corresponds to an area of 0.08 to the left of the distribution. Using a standard normal (z) table or statistical software or calculator, we find that \(z(0.08)\) is approximately -1.41.
02

Calculate the area between \(z(0.98)\) and \(z(0.02)\)

This is asking for the area under the curve between the z-scores that correspond to the areas 0.98 and 0.02. \[z(0.98)\approx 2.05\] \[z(0.02)\approx -2.05\] The difference between two areas under the curve: \[0.98 - 0.02 = 0.96\], so the area between \(z(0.98)\) and \(z(0.02)\) is 0.96.
03

Find the Z-score for \(z(1.00-0.01)\)

We need to find the z-score for the area 0.99 to the left of the distribution. From z-table or statistical software, we can find that \(z(0.99)\) is approximately 2.33.
04

Calculate the Z-score difference \(z(0.025)-z(0.975)\)

We are asked to find the difference between z-scores corresponding to areas 0.975 and 0.025. \[z(0.975)\approx 1.96\] \[z(0.025)\approx -1.96\] So their difference would be:\[ 1.96 - (-1.96) = 3.92 \]

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.

Standard Normal Distribution
The standard normal distribution is a crucial concept in statistics representing a special case of the normal distribution. It's characterized by a mean of 0 and a standard deviation of 1. This bell-shaped curve is symmetrical, with the majority of data points falling within three standard deviations from the mean.

When dealing with z-scores, we are referring to the number of standard deviations a data point is from the mean. For instance, a z-score of 1.0 implies that the data point is one standard deviation above the mean. In practice, this allows us to compare individual scores across different normal distributions by converting them to a common scale, the standard normal distribution.

Understanding the z-score is fundamental for interpreting the area under the curve. For example, in the provided exercise, z(0.08) indicates the z-score with an area of 0.08 to the left under the standard normal curve. By consulting standard normal distribution tables, or using software, this area corresponds to a z-score, which in this case is approximately -1.41.
Area Under the Curve
The area under the curve in a standard normal distribution represents the probability or proportion of the data within a certain range. Since the total area under the curve equals 1 or 100%, we can interpret specific areas as the likelihood of a data point falling within a specific range.

Let's relate this to the exercise. The area between z(0.98) and z(0.02) requires finding the z-scores first. Z(0.98) is about 2.05, z(0.02) is about -2.05, and their difference reflects the range of data points between these two z-scores. With 0.98 to the right and 0.02 to the left, we subtract these to find the area between them, which is 0.96 or 96%. This significant area includes the majority of data and aligns with the bell-shaped curve's characteristic that most data points lie close to the mean.

To improve understanding, visual aids such as diagrams depicting these areas can greatly enhance comprehension for students navigating the concept of area under the standard normal curve.
Normal Distribution Tables
Normal distribution tables, also known as z-tables, are a practical tool for finding areas under the standard normal curve or determining z-scores. These tables provide the area to the left of a z-score in a standard normal distribution. To use these tables, you locate the z-score of interest, and corresponding to it, you'll find the area or probability.

In our original exercise, we use a z-table to find the z-scores corresponding to given areas. For example, z(1.00-0.01) or z(0.99) corresponds to a z-score of approximately 2.33. Similarly, for the difference in z-scores in part d of the exercise, we use the table to find z(0.975) and z(0.025), which is about 1.96 and -1.96, respectively. Their difference, 3.92, is simply the distance between those z-scores on the standard normal distribution.

While these tables may seem daunting at first, they are a fundamental part of interpreting standard normal distributions. Combining instructions with visuals showing how to read these tables effectively could significantly improve ease of use and understanding for students.

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 following areas under the standard normal curve. a. To the right of \(z=3.18, P(z>3.18)\) b. To the right of \(z=1.84, P(z>1.84)\) c. To the right of \(z=0.75, P(z>0.75)\)

The grades on an examination whose mean is 525 and whose standard deviation is 80 are normally distributed. a. Anyone who scores below 350 will be retested. What percentage does this represent? b. The top \(12 \%\) are to receive a special commendation. What score must be surpassed to receive this special commendation? c. The interquartile range of a distribution is the difference between \(Q_{1}\) and \(Q_{3}, Q_{3}-Q_{1}\). Find the interquartile range for the grades on this examination. d. Find the grade such that only 1 out of 500 will score above it.

Not all NBA coaches who enjoy lengthy careers consistently put together winning seasons with the teams they coach. For example, Bill Fitch, who coached 25 seasons of professional basketball after starting his coaching career at the University of Minnesota, won 944 games but lost 1106 while working with the Cavaliers, Celtics, Rockets, Nets, and Clippers. If you were to randomly select 60 box scores from the historical records of games in which Bill Fitch coached one of the teams, what is the probability that fewer than half of them show his team winning? To obtain your answer, use the normal approximation to the binomial distribution.

According to a December 2008 report from the Join Together website of the Boston University School of Public Health, approximately half (42\%) of U.S. children are exposed to secondhand smoke on a weekly basis, with more than \(25 \%\) of parents reporting that their child has been exposed to smoke in their homes. This statistic was one of many results from the Social Climate Survey of Tobacco Control [http://www.socialclimate.org/]. Use the normal approximation to the binomial distribution to find the probability that in a poll of 1200 randomly selected parents, between 450 and 500 inclusive will report that their child has been exposed to smoke on a weekly basis. a. Solve using normal approximation and Table 3 in Appendix B. b. Solve using a computer or calculator and the normal approximation method. c. Solve using a computer or calculator and the binomial probability function.

As shown in Example \(6.12,\) incomes of junior executives are normally distributed with a standard deviation of \(\$ 3828.\) a. What is the mean for the salaries of junior executives, if a salary of \(\$ 62,900\) is at the top end of the middle \(80 \%\) of incomes? b. With the additional information learned in part a, what is the probability that a randomly selected junior executive earns less than \(\$ 50,000 ?\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.