/*! 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 20 Sketch the areas under the stand... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 20

Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas. To the right of \(z=0.15\)

Short Answer

Expert verified
The area to the right of \(z=0.15\) is approximately 0.4404.

Step by step solution

01

Understand the Standard Normal Curve

The standard normal distribution is a bell-shaped distribution with a mean of 0 and a standard deviation of 1. The area under the curve represents probabilities or percentages.
02

Identify the Given Interval

We need to find the area under the standard normal curve to the right of \(z=0.15\). This area represents the probability that a value is greater than \(z=0.15\).
03

Use the Standard Normal Table or Z-Score Calculator

A standard normal (Z) table or a Z-score calculator is typically used to find the probabilities related to Z-scores. Look up \(z=0.15\) in the Z-table to find the area to the left of \(z=0.15\). This gives you the cumulative probability.
04

Calculate the Complementary Area

The cumulative area to the left of \(z=0.15\) from the Z-table is approximately 0.5596. To find the area to the right, subtract this value from 1: \(1 - 0.5596 = 0.4404\).
05

Sketch the Curve

Sketch the standard normal curve, mark \(z=0.15\) on the horizontal axis, and shade the area to the right of this point. This shaded area represents 0.4404 or 44.04% 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.

Standard Normal Distribution
The standard normal distribution is a foundational concept in statistics. It's a specific continuous probability distribution characterized by a symmetric, bell-shaped curve. Its most unique feature is its mean of 0 and a standard deviation of 1.

This type of distribution is important because it's a model for various natural phenomena and standardized datasets. When a dataset follows this distribution, it allows statisticians and researchers to make various predictions and analyses.
  • **Mean**: The average, centered at 0.
  • **Standard deviation**: Measures dispersion, standardized to 1.
  • **Symmetry**: The curve is symmetrical about the mean.
  • **Total Area**: Equals 1, representing 100% of the probabilities.
Understanding this distribution helps in simplifying the calculation of probabilities and making inferences about datasets.
Z-Scores
Z-scores are essential when working with the standard normal distribution. A Z-score measures how many standard deviations an element is from the mean of the distribution.

This transformation makes it easier to compare different data points from various datasets. The formula for calculating a Z-score is:\[ Z = \frac{(X - \mu)}{\sigma} \]
  • \(X\) is the raw score or data point.
  • \(\mu\) is the mean of the distribution.
  • \(\sigma\) is the standard deviation of the distribution.
By converting scores to Z-scores, you put them on a common scale, which is particularly helpful when using the standard normal (Z) table.
Probability Calculations
Probability calculations using the standard normal distribution often involve finding areas under the curve associated with Z-scores. These areas correspond to probabilities.

To find a probability using a Z-table:- First, look up the cumulative probability, which is the area to the left of the Z-score. - If you need the area to the right, which is often the case, compute it using the complement rule: \[ \text{Right area} = 1 - \text{Left area} \]In our step-by-step solution, we calculated \[1 - 0.5596 = 0.4404 \]This means there's a 44.04% chance of obtaining a value greater than \(z=0.15\). These calculations are vital for hypothesis testing and statistical inference.
Standard Normal Curve
The standard normal curve graphically represents the standardized normal distribution, making it an integral part of visualization in statistics.

Using this curve:
  • **Visualize data**: Easy identification of probabilities.
  • **Interpret**: Clear depiction of how data points relate to the mean.
  • **Communicate**: Simplified presentation of statistical concepts.
The curve's symmetry makes it easy to estimate probabilities visually. For instance, marking \(z=0.15\) on the curve and shading the area to the right lets you intuitively understand the 44.04% probability related to this Z-score. Visualization through the curve is not just helpful; it's fundamental for students and professionals alike, ensuring a more intuitive grasp of complex statistical methodologies.

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

Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 28 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 5 months, and the distribution of lifetimes is normal. (a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production should the company expect to replace? (b) If Accrotime does not want to make refunds on more than \(12 \%\) of the watches it makes, how long should the guarantee period be (to the nearest month)?

Let \(x\) be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12 -hour fast. Assume that for people under 50 years old, \(x\) has a distribution that is approximately normal, with mean \(\mu=85\) and estimated standard deviation \(\sigma=25\) (based on information from Diagnostic Tests with Nursing Applications, edited by S. Loeb, Springhouse). A test result \(x<40\) is an indication of severe excess insulin, and medication is usually prescribed. (a) What is the probability that, on a single test, \(x<40\) ? (b) Suppose a doctor uses the average \(\bar{x}\) for two tests taken about a week apart. What can we say about the probability distribution of \(\bar{x}\) ? Hint: See Theorem 6.1. What is the probability that \(\bar{x}<40\) ? (c) Repeat part (b) for \(n=3\) tests taken a week apart. (d) Repeat part (b) for \(n=5\) tests taken a week apart. (e) Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as \(n\) increased? Explain what this might imply if you were a doctor or a nurse. If a patient had a test result of \(\bar{x}<40\) based on five tests, explain why either you are looking at an extremely rare event or (more likely) the person has a case of excess insulin.

Distribution Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us. (a) Suppose \(n=33\) and \(p=0.21\). Can we approximate the \(\hat{p}\) distribution by a normal distribution? Why? What are the values of \(\mu_{\hat{p}}\) and \(\sigma_{\hat{p}} ?\) (b) Suppose \(n=25\) and \(p=0.15 .\) Can we safely approximate the \(\hat{p}\) distribution by a normal distribution? Why or why not? (c) Suppose \(n=48\) and \(p=0.15 .\) Can we approximate the \(\hat{p}\) distribution by a normal distribution? Why? What are the values of \(\mu_{\hat{p}}\) and \(\sigma_{\hat{p}} ?\)

Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(3 \leq x \leq 6) ; \mu=4 ; \sigma=2 $$

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean \(\mu=75\) tons and standard deviation \(\sigma=0.8\) ton. (a) What is the probability that one car chosen at random will have less than \(74.5\) tons of coal? (b) What is the probability that 20 cars chosen at random will have a mean load weight \(\bar{x}\) of less than \(74.5\) tons of coal? (c) Suppose the weight of coal in one car was less than \(74.5\) tons. Would that fact make you suspect that the loader had slipped out of adjustment? Suppose the weight of coal in 20 cars selected at random had an average \(\bar{x}\) of less than \(74.5\) tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

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.