/*! 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 29 Draw a normal curve and label th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 29

Draw a normal curve and label the mean and inflection points. $$ \mu=30 \text { and } \sigma=10 $$

Short Answer

Expert verified
The mean is at 30, and the inflection points are at 20 and 40.

Step by step solution

01

- Draw the Horizontal Axis

Start by drawing a horizontal axis on your graph. This axis will represent the variable for which we are plotting the normal distribution.
02

- Plot the Mean

Identify the mean of the distribution, denoted as \( \mu = 30 \). Mark this point on the horizontal axis.
03

- Draw the Bell Curve

Sketch a symmetric bell-shaped curve centered on the mean (\( \mu = 30 \)). Ensure the curve is smooth and approaches the horizontal axis asymptotically.
04

- Identify and Mark the Inflection Points

The inflection points occur at \( \mu - \sigma \) and \( \mu + \sigma \). Calculate these points: \( 30 - 10 = 20 \) and \( 30 + 10 = 40 \). Mark these points on the horizontal axis.
05

- Label the Graph

Label the mean (\( \mu = 30 \)) and the inflection points (\(20\) and \(40\)) clearly on the graph.

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.

mean
The mean is the average value of a data set. In a normal distribution, it is represented by the symbol \(\mu\). The mean is the center point of a bell curve.
Here鈥檚 how you can find and plot the mean on a graph:
  • Start with a horizontal axis.
  • Identify the mean value; in this exercise, it is given as \(\mu = 30\).
  • Mark this point on the horizontal axis. This will be the center of your bell curve.
By plotting the mean, you provide a reference point that helps in identifying the shape and center of the normal distribution.
inflection points
Inflection points are where the curve changes concavity. For a normal distribution, these points are crucial because they indicate the spread of the data.
To locate inflection points:
  • Identify the mean \(\mu\) and standard deviation \(\sigma\). In this exercise, \(\mu = 30\) and \(\sigma = 10\).
  • Calculate the positions: \(\mu - \sigma = 20\) and \(\mu + \sigma = 40\).
  • Mark these points on the horizontal axis.
These points divide the bell curve into sections where the curve transitions from being concave upwards to concave downwards and vice versa. This visual change helps you understand the distribution's variability.
standard deviation
Standard deviation is a measure of spread in your dataset. Represented by \(\sigma\), it indicates how much individual data points deviate from the mean.
A few key aspects to understand about standard deviation in a normal distribution:
  • It determines the width of the bell curve.
  • The smaller the standard deviation, the steeper and narrower the curve.
  • Larger standard deviations result in flatter and wider curves.
In this exercise, the standard deviation is \(10\). It helps in finding the inflection points, which are crucial for understanding the spread and shape of the distribution.
bell curve
A bell curve, or normal distribution curve, is a graphical representation of data that shows a symmetric, bell-shaped appearance. Here are its key features:
  • The mean (\(\mu\)) is at the center of the curve.
  • Inflection points are located one standard deviation (\(\sigma\)) away from the mean.
  • The curve approaches, but never touches the horizontal axis.
In this exercise, the bell curve is centered at the mean, which is 30. The curve is smoothly drawn so that it has two equal halves about this central point. This distribution is used in statistics to represent real-world data that clusters around an average value. Understanding the bell curve allows you to grasp concepts like probability and data variability easily.

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

Monthly charges for cell phone plans in the United States are normally distributed with mean \(\mu=\$ 62\) and standard deviation \(\sigma=\$ 18 .\) (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of plans that charge less than \(\$ 44\) (c) Suppose the area under the normal curve to the left of \(x=\$ 44\) is 0.1587 . Provide two interpretations of this result.

In a recent poll, the Gallup Organization found that \(45 \%\) of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 500 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. Use the normal approximation to the binomial to approximate the probability that (a) exactly 250 of those surveyed feel the state of morals is poor. (b) no more than 220 of those surveyed feel the state of morals is poor. (c) more than 250 of those surveyed feel the state of morals is poor. (d) between 220 and 250 , inclusive, believe the state of morals is poor. (e) at least 260 adult Americans believe the overall state of moral values is poor. Would you find this result unusual? Why?

Find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies to the left of (a) \(z=-3.49\) (b) \(z=-1.99\) (c) \(z=0.92\) (d) \(z=2.90\)

Assume that the random variable \(X\) is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7 .\) Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(38

The lengths of human pregnancies are approximately normally distributed, with mean \(\mu=266\) days and standard deviation \(\sigma=16\) days. (a) What proportion of pregnancies lasts more than 270 days? (b) What proportion of pregnancies lasts less than 250 days? (c) What proportion of pregnancies lasts between 240 and 280 days? (d) What is the probability that a randomly selected pregnancy lasts more than 280 days? (e) What is the probability that a randomly selected pregnancy lasts no more than 245 days? (f) A "very preterm" baby is one whose gestation period is less than 224 days. Are very preterm babies unusual?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics 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.