/*! 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 23 Sketch the areas under the stand... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 23

Sketch the areas under the standard normal curve over the indicated intervals, and find the specified areas. Between \(z=0\) and \(z=3.18\)

Short Answer

Expert verified
The area between \(z=0\) and \(z=3.18\) is 0.4993.

Step by step solution

01

Understand the Standard Normal Curve

The standard normal curve is a bell-shaped curve that is symmetric about the mean, which is 0. It represents the distribution of a standard normal random variable with a mean of 0 and a standard deviation of 1.
02

Identify the Interval

We need to find the area under the standard normal curve between the values of \(z=0\) and \(z=3.18\). This means we are interested in the probability that a standard normal random variable falls between these two values.
03

Use the Standard Normal Distribution Table

The standard normal distribution table (or Z-table) provides the area to the left of a specific z-value. First, look up the area to the left of \(z=3.18\). According to the Z-table, this value is approximately 0.9993.
04

Determine the Area to the Left of \(z=0\)

Since \(z=0\) is the mean of the standard normal distribution, the area to the left of \(z=0\) is exactly 0.5.
05

Calculate the Area Between the Two Z-values

To find the area between \(z=0\) and \(z=3.18\), subtract the area to the left of \(z=0\) from the area to the left of \(z=3.18\). Calculate: \[\text{Area} = 0.9993 - 0.5 = 0.4993\] This is the probability of a standard normal random variable being between 0 and 3.18.

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.

Z-table
The Z-table, also known as the standard normal distribution table, is a crucial tool in statistics. It helps us find the area under the normal curve for any given z-value. This area represents the probability that a standard normal random variable will be less than or equal to that z-value.
Imagine you have a number line of z-values, and you want to know how likely it is for a value to fall at random up to a specific z-point.
  • To use the Z-table effectively:
    • Locate your z-value in the left column (which shows the z-value to the single decimal place).
    • Move across the row to find the matching value in the column header indicating the second decimal point of your z-value.
    • The intersection provides the probability or the area under the curve to the left of that z-point.
    Using a Z-table simplifies finding these probabilities, making it broadly applicable in fields like finance and psychology.
    Probability
    Probability is all about measuring the likelihood of an event happening. When working with the standard normal distribution, probability refers to the area under the curve between two z-values.
    Let's break it down:
    • Each portion of the curve tells us how likely it is for a certain range of values to occur.
    • For the standard normal distribution, any z-value corresponds to a specific probability, which we can find using the Z-table.
    • For example, when calculating the area between two z-values, like from 0 to 3.18, you're essentially measuring the probability of a variable falling within that range.
    This makes probability handy in practical scenarios. Imagine estimating the chance of sales exceeding a certain threshold or the likelihood of a measurement falling within a specified range.
    Normal Curve
    The normal curve, often called the bell curve due to its shape, is a foundational element in statistics. It reflects how many natural phenomena distribute, with most occurring around the central mean and fewer happening further from it.
    • The standard normal curve centers on a mean of 0, with a standard deviation of 1, defining how spread out the values are.
    • It is symmetric, meaning both sides of the mean are mirror images.
    • Under this curve, the total area is equal to 1, representing the entire range of probabilities.
    A fundamental property of the curve is that specific z-values tell us how many standard deviations away from the mean a point is. Understanding and visualizing this curve enables statisticians and data analysts to assess data patterns and make predictions that are crucial for data-driven decisions.

    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

    What is a sampling distribution?

    Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(7 \leq x \leq 9) ; \mu=5 ; \sigma=1.2 $$

    Consider an \(x\) distribution with standard deviation \(\sigma=12\). (a) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 2, how large does the sample size need to be? (b) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 1, how large does the sample size need to be?

    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. $$ P(-2.18 \leq z \leq-0.42) $$

    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

    Applied Mathematics

    Read Explanation

    Theoretical and Mathematical Physics

    Read Explanation

    Probability and Statistics

    Read Explanation

    Decision Maths

    Read Explanation

    Mechanics Maths

    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.