/*! 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 3 True or False: The normal curve ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 3

True or False: The normal curve is symmetric about its mean, \(\mu .\)

Short Answer

Expert verified
True

Step by step solution

01

Understanding the Normal Curve

The normal curve, also known as the Gaussian distribution or bell curve, is a continuous probability distribution characterized by its symmetrical bell shape.
02

Symmetry of the Curve

A curve is symmetric if one half of it is a mirror image of the other half. For the normal distribution, the peak of the curve, which is the mean ( 渭 ), divides the curve into two identical halves.
03

Implications of Symmetry

Since the normal distribution is symmetric about the mean, any value below the mean has a corresponding value above the mean with the same probability.
04

Conclusion

Given that the normal curve is symmetric about its mean, the statement 'The normal curve is symmetric about its mean, 渭 ' is true.

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.

Gaussian distribution
The Gaussian distribution is another name for the normal distribution. It describes how values are distributed in many real-world situations. This distribution is continuous, meaning it includes every possible value within a certain range rather than just specific points.
Gaussian distribution is defined by two main parameters: the mean (渭) and the standard deviation (蟽). The mean is the center of the distribution, where most values cluster, while the standard deviation measures how spread out the values are around the mean.
Understanding this distribution helps in predicting probabilities and outcomes within a data set. It's widely used in fields like statistics, finance, and natural sciences due to its natural occurrence in varied phenomena.
bell curve
The bell curve is a name given to the shape of the normal distribution, which looks like a bell. This visual representation helps in understanding the distribution鈥檚 properties.
In a bell curve, most data points gather around the mean, creating a peak. As you move away from the mean, the frequency of data points gradually decreases on both sides, forming the downwards slopes of the bell.
Key characteristics:
  • Symmetry: The bell curve is symmetric around the mean. Each half is a mirror image of the other.
  • Unimodal: There is only one peak.
  • Asymptotic: The tails of the curve get closer to the horizontal axis but never touch it.
This symmetry is crucial as it implies consistent frequencies for values equally distant from the mean.
mean symmetry
Mean symmetry is a critical property of the normal distribution. It implies that the distribution is evenly balanced around the mean value.
If you were to fold the normal curve at the mean, each side would align perfectly with the other. This symmetry indicates that any deviation below the mean is mirrored by an equivalent deviation above the mean with equal probability.
Understanding mean symmetry can simplify the analysis of data. Since both sides are identical, you often only need to study one side to make inferences about the entire distribution. Recognizing this can help in predicting behaviors and outcomes based on the given data.
This property is the basis for many statistical methods and concepts, making it essential to grasp fully.

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

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. \(P(X>65)\)

According to the Current Population Survey (Internet release date: September 15,2004\(), 55 \%\) of males between the ages of 18 and 24 years lived at home in \(2003 .\) (Unmarried college students living in a dorm are counted as living at home.) Suppose a survey is administered today to 200 randomly selected males between the ages of 18 and 24 years. and 130 of them respond that they live at home. (a) Approximate the probability that such a survey will result in at least 130 of the respondents living at home under the assumption that the true percentage is \(55 \%\). (b) What does the result from part (a) suggest?

Steel rods are manufactured with a mean length of 25 centimeters \((\mathrm{cm}) .\) Because of variability in the manufacturing process, the lengths of the rods are approximately normally distributed, with a standard deviation of \(0.07 \mathrm{~cm} .\) (a) What proportion of rods has a length less than \(24.9 \mathrm{~cm} ?\) (b) Any rods that are shorter than \(24.85 \mathrm{~cm}\) or longer than \(25.15 \mathrm{~cm}\) are discarded. What proportion of rods will be discarded? (c) Using the results of part (b), if 5000 rods are manufactured in a day, how many should the plant manager expect to discard? (d) If an order comes in for 10,000 steel rods, how many rods should the plant manager manufacture if the order states that all rods must be between \(24.9 \mathrm{~cm}\) and \(25.1 \mathrm{~cm} ?\)

Researchers conducted a prospective cohort study in which male patients who had an out-of-hospital cardiac arrest were submitted to therapeutic hypothermia (intravenous infusion of cold saline followed by surface cooling with the goal of maintaining body temperature of 33 degrees Celsius for 24 hours. Note that normal body temperature is 37 degrees Celsius). The survival status, length of stay in the intensive care unit (ICU), and time spent on a ventilator were measured. Each of these variables was compared to a historical cohort of patients who were treated prior to the availability of therapeutic hypothermia. Of the 52 hypothermia patients, 37 survived; of the 74 patients in the control group, 43 survived. The median length of stay among survivors for the hypothermia patients was 14 days versus 21 days for the control group. The time on the ventilator among survivors for the hypothermia group was 219 hours versus 328 hours for the control group. (a) What does it mean to say this is a prospective cohort study? (b) What is the explanatory variable in the study? Is it qualitative or quantitative? (c) What are the three response variables in the study? For each, state whether the variable is qualitative or quantitative. (d) Is time on the ventilator a statistic or parameter? Explain. (e) To what population does this study apply? (f) Based on the results of this study, what is the probability a randomly selected male who has an out-of-hospital cardiac arrest and submits to therapeutic hypothermia wil survive? What about those who do not submit to therapeutic hypothermia?

Draw a normal curve and label the mean and inflection points. $$ \mu=50 \text { and } \sigma=5 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure 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.