/*! 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 15 Find the \(z\) value described a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 15

Find the \(z\) value described and sketch the area described. Find \(z\) such that \(6 \%\) of the standard normal curve lies to the left of \(z\).

Short Answer

Expert verified
The z-value is approximately \(-1.555\), with the shaded area to the left of this point representing 6\%.

Step by step solution

01

Understanding the Standard Normal Curve

A standard normal curve is a special normal distribution where the mean is 0 and the standard deviation is 1. We need to find the value of \(z\) such that the area to the left of \(z\) is 0.06, meaning only 6\% of the data falls below that point.
02

Using the Z-Table or Inverse Normal Function

To find the \(z\)-value corresponding to an area of 0.06, we can use a standard normal distribution table (z-table) or a calculator with an inverse normal function (often denoted as \(\text{invNorm}\)). Search for the area closest to 0.06 or input 0.06 into the inverse normal function.
03

Locate the Z-Value

Upon using the z-table or inverse normal calculation, we find that the closest area to 0.06 corresponds to a \(z\)-value of approximately \(-1.555\). This means 6\% of the data in a standard normal distribution lies to the left of \(z = -1.555\).
04

Sketching the Area on the Curve

To sketch the area, draw a standard normal curve with a mean at 0. Then, mark the \(z\)-value of \(-1.555\) on the horizontal axis, to the left of the mean. Shade the area to the left of \(z = -1.555\) to represent the 6\% of the total area under the curve.

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.

Understanding Z-Score
The Z-score is a way of standardizing variables so that they can be compared and interpreted more easily. It's essentially a measure of how far away a particular data point is from the mean, in terms of standard deviations. For a standard normal distribution, the Z-score tells us how many standard deviations an element is from the mean, which is typically set at 0.
  • A positive Z-score indicates the value is above the mean.
  • A negative Z-score shows the value is below the mean.
  • Z-scores help in identifying outliers or extreme values in data sets.
By calculating Z-scores, we can determine probabilities and make data comparisons more comprehensible. This becomes particularly helpful when working with normal distributions.
Inverse Normal Function Explained
The inverse normal function is a statistical function used to find the Z-score that corresponds to a given cumulative probability in a standard normal distribution. In other words, it's the opposite of finding probabilities for a Z-score, as you start with the probability and find the associated Z-score instead.

This function is available in statistical calculators and software, often labeled as "invNorm." You simply input the probability to find the Z-score.
  • It helps determine cut-off points in a data set.
  • Useful in hypothesis testing and confidence intervals.
  • Convenient for interpreting results without extensive manual calculations.
By using this function, students and researchers can directly find the critical values needed for analysis without consulting extensive tables.
Using Z-Table for Standard Normal Distribution
The Z-table, also known as the standard normal table, is a mathematical table used to find the probability of a statistic falling below a given Z-score in a standard normal distribution. Each entry in the table corresponds to an area under the standard normal curve to the left of a particular Z-score.

Using the Z-table, you can easily find:
  • Cumulative probabilities associated with Z-scores.
  • The Z-score for a specified cumulative probability.
  • Critical values for hypothesis testing.
The table is designed for ease of use. You simply locate the row for the Z-score鈥檚 first two digits and the column for the second decimal. This is especially useful when a calculator or computer isn鈥檛 available.
Visualizing the Normal Distribution Curve
The normal distribution curve is a symmetrical, bell-shaped graph that represents the distribution of many types of data. Most points cluster around the mean, and probabilities taper off symmetrically as they move away from the center.
  • The peak of the curve represents the mean, median, and mode of the distribution.
  • The spread is determined by the standard deviation, which indicates dispersion around the mean.
  • Understanding this curve helps in grasping the concept of variability within datasets.
In our exercise, by sketching the section of the curve where 6% of values lie to the left of a given Z-score, we gain a visual understanding of what this probability means in real-world data. This visualization is a powerful tool for those studying statistics and data analysis, as it makes theoretical concepts more intuitive.

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

Suppose \(x\) has a distribution with a mean of 20 and a standard deviation of \(3 .\) Random samples of size \(n=36\) are drawn. (a) Describe the \(\bar{x}\) distribution and compute the mean and standard deviation of the distribution. (b) Find the \(z\) value corresponding to \(\bar{x}=19\). (c) Find \(P(\bar{x}<19)\). (d) Would it be unusual for a random sample of size 36 from the \(x\) distribution to have a sample mean less than 19? Explain.

The amount of money spent weekly on cleaning, maintenance, and repairs at a large restaurant was observed over a long period of time to be approximately normally distributed, with mean \(\mu=\$ 615\) and standard deviation \(\sigma=\$ 42\). (a) If \(\$ 646\) is budgeted for next week, what is the probability that the actual costs will exceed the budgeted amount? (b) How much should be budgeted for weekly repairs, cleaning, and maintenance so that the probability that the budgeted amount will be exceeded in a given week is only 0.10?

Binomial probability distributions depend on the number of trials \(n\) of a binomial experiment and the probability of success \(p\) on each trial. Under what conditions is it appropriate to use a normal approximation to the binomial?

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of \(5.1\) millimeters \((\mathrm{mm})\) and a standard deviation of \(0.9 \mathrm{~mm}\) (Source: Homol'ovi II: Archaeology of an Ancestral Hopi Village, Arizona, edited by E. C. Adams and K. A. Hays, University of Arizona Press). For a randomly found shard, what is the probability that the thickness is (a) less than \(3.0 \mathrm{~mm}\) ? (b) more than \(7.0 \mathrm{~mm}\) ? (c) between \(3.0 \mathrm{~mm}\) and \(7.0 \mathrm{~mm}\) ?

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-2.20 \leq z \leq 1.40) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.