/*! 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 8 List the major characteristics o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 8

List the major characteristics of a normal probability distribution.

Short Answer

Expert verified
A normal distribution is symmetric, bell-shaped, with mean = median = mode, described by its standard deviation.

Step by step solution

01

Understanding the Shape

A normal probability distribution is commonly described as bell-shaped and symmetric around the mean. This means that the left and right halves of the graph are mirror images of each other, showing that the data is evenly distributed around the center point, which is the mean.
02

Mean, Median, and Mode Alignment

In a normal distribution, the mean, median, and mode are all the same. They all occur at the peak of the distribution and divide the data into two equal halves, underscoring the symmetry of the distribution.
03

Describing the Spread with Standard Deviation

The spread or dispersion of the data in a normal distribution is determined by the standard deviation. Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations. This is known as the Empirical Rule.
04

Probability Density Function

The probability density function (PDF) of a normal distribution is given by a specific mathematical equation: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]where \( \mu \) is the mean and \( \sigma \) is the standard deviation. This function describes the probability that a variable takes a specific value.
05

Tails Behavior

In a normal distribution, the tails extend to infinity in both directions but never actually touch the horizontal axis. This indicates that while extreme values are rare, they are possible, and the probability never reaches zero.

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.

Probability Density Function
The Probability Density Function (PDF) is a crucial concept in understanding normal distribution. It is represented by a precise mathematical formula: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]Here, \( \mu \) symbolizes the mean, and \( \sigma \) is the standard deviation.
This function provides a smooth curve (the bell shape) over a range of values, predicting how likely it is for a variable to assume a particular value.
  • The PDF never goes negative and only reaches zero at infinity, indicating that extreme values are improbable but possible.
  • The area under the curve between any two points is the probability that the variable falls within that range.
Using the PDF, we can visualize how probabilities are distributed across different values, centered around the mean with probabilities tapering off symmetrically as we move away from it.
Empirical Rule
The Empirical Rule is an easy way to grasp data dispersion in a normal distribution using the standard deviation. This rule states that, in a normal distribution, most of the data lies close to the mean:
  • Approximately 68% of the data falls within one standard deviation of the mean.
  • About 95% lies within two standard deviations.
  • Nearly 99.7% is within three standard deviations.
This predictable pattern allows us to make informed guesses about the probability of data points.
It helps describe how data clusters around the mean, highlighting the significance of standard deviation in understanding distribution.
Standard Deviation
Standard Deviation is a key measure of variability or spread in a data set. In the normal distribution:
1. It tells us how much the data deviates from the mean. 2. A small standard deviation means data points are close to the mean. 3. A large standard deviation signifies data widely spread around the mean.
Using standard deviation:
  • We evaluate the consistency of the data set.
  • It aids in comparing variability between different data sets.
For example, knowing that 68% of the data is within one standard deviation makes it a powerful tool to predict outcomes within a known range.
In practice, understanding standard deviation helps analyze risks and variations effectively in various fields like finance, research, and quality control.

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

Explain what is meant by this statement: "There is not just one normal probability distribution but a 'family' of them."

A uniform distribution is defined over the interval from 6 to 10. a. What are the values for \(a\) and \(b\) ? b. What is the mean of this uniform distribution? c. What is the standard deviation? d. Show that the total area is 1.00 . e. Find the probability of a value more than 7 . f. Find the probability of a value between 7 and \(9 .\)

A normal distribution has a mean of 50 and a standard deviation of 4. a. Compute the probability of a value between 44.0 and 55.0 . b. Compute the probability of a value greater than 55.0 . c. Compute the probability of a value between 52.0 and 55.0 .

According to the South Dakota Department of Health, the number of hours of TV viewing per week is higher among adult women than adult men. A recent study showed women spent an average of 34 hours per week watching TV and men, 29 hours per week. Assume that the distribution of hours watched follows the normal distribution for both groups and that the standard deviation among the women is 4.5 hours and is 5.1 hours for the men. a. What percent of the women watch TV less than 40 hours per week? b. What percent of the men watch TV more than 25 hours per week? c. How many hours of TV do the \(1 \%\) of women who watch the most TV per week watch? Find the comparable value for the men.

Best Electronics Inc. offers a "no hassle" returns policy. The daily number of customers returning items follows the normal distribution. The mean number of customers returning items is 10.3 per day and the standard deviation is 2.25 per day. a. For any day, what is the probability that eight or fewer customers returned items? b. For any day, what is the probability that the number of customers returning items is between 12 and \(14 ?\) c. Is there any chance of a day with no customer returns?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.