/*! 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 22 The number of hours a battery la... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 22

The number of hours a battery lasts before failing varies from battery to battery. The distribution of failure times follows an exponential distribution (see Example \(15.7\) ), which is strongly skewed to the right. The central limit theorem says that (a) as we look at more and more batteries, their average failure time gets ever closer to the mean \(\mu\) for all batteries of this type. (b) the average failure time of a large number of batteries has a distribution of the same shape (strongly skewed) as the distribution for individual batteries. (c) the average failure time of a large number of batteries has a distribution that is close to Normal.

Short Answer

Expert verified
The correct answer is (c).

Step by step solution

01

Understanding the Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided the samples are independent and identically distributed.
02

Identify Potential Choice

Given the choices: (a) mentions average getting closer to true mean, which is part of CLT's implications but not about distribution shape. (b) says the shape remains skewed, contradicting the CLT. (c) suggests that the distribution of the sample mean becomes normal, aligning with CLT principles.
03

Evaluate the Choices

(a) is partially correct in the sense of individual averages approaching the mean, but doesn't address shape. (b) is incorrect because the CLT specifically says the sample mean's distribution approaches normal, not remaining skewed. (c) accurately captures that the CLT predicts the sample mean鈥檚 distribution becomes close to Normal.
04

Conclusion Based on CLT

The Central Limit Theorem directly supports choice (c): "the average failure time of a large number of batteries has a distribution that is close to Normal." This is the correct effect of the CLT on the shape of the distribution of averages.

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.

Exponential Distribution
An exponential distribution is a probability distribution that describes the time between consecutive events in a process that occurs continuously and independently at a constant average rate. It's commonly used to model situations where there is a constant probability of an event occurring over time, such as the failure times of electronic components or the lifespan of organisms. One of the key characteristics of the exponential distribution is its high degree of skewness.

  • The exponential distribution is always skewed to the right. This means that the majority of the data points are concentrated on the left of the graph, with a long "tail" stretching out to the right.
  • It is defined by one parameter, the rate \( \lambda \), which is the reciprocal of the mean \( \mu \). The mean is given by \( \mu = \frac{1}{\lambda} \).
  • The variance of the exponential distribution is given by \( \frac{1}{\lambda^2} \).
Understanding these properties allows you to predict behavior over time, such as how long until a machine breaks down.
Sample Mean Distribution
The sample mean distribution refers to the distribution of the average values drawn from random samples of a population. When you repeatedly draw samples and calculate their means, you create a distribution of those means. This is a fundamental concept in statistics, underpinning much of inferential statistics.

  • According to the Central Limit Theorem, the sample mean distribution will be approximately normal if the sample size is sufficiently large, regardless of the shape of the population distribution.
  • The mean of the sample mean distribution equals the population mean \( \mu \).
  • The standard deviation of the sample mean distribution, also known as the standard error, is given by \( \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size.
This allows statisticians to make predictions about population parameters based on sample data.
Normal Distribution
Normal distribution, often referred to as a bell curve, is a probability distribution that is symmetric about the mean. It describes data that clusters around a central bound or the average, and is widespread in nature and statistics.

  • A normal distribution has two key parameters: the mean \( \mu \), which determines the center of the graph, and the standard deviation \( \sigma \), which determines the width of the graph.
  • Because of its bell shape, most values lie close to the mean with fewer values trailing off as you move away from the center.
  • The Central Limit Theorem ensures that, under certain conditions, sums or averages of random variables will tend to become normally distributed, even if the original variables themselves are not normally distributed.
Understanding normal distribution is crucial because it forms the basis of many statistical methods and hypothesis tests.
Skewness
Skewness measures the asymmetry of a distribution. If a distribution is symmetrical, its skewness is zero. However, distributions can lean heavily to the left or right, causing a skew. Understanding skewness helps in comprehending the distribution's behavior and the potential effect of outliers.

  • A right-skewed (or positively skewed) distribution has a longer tail on the right. This often indicates where data points are less frequent.
  • A left-skewed (or negatively skewed) distribution features a longer tail on the left.
  • Exponential distributions are examples of right-skewed distributions, while an example of a left-skewed distribution could be exam scores that are very challenging.
Recognizing skewness in data is important for selecting appropriate statistical methods.
Distribution Shape
The shape of a distribution is a fundamental aspect of its nature, revealing the underlying characteristics of the data set. Shapes can be normal, skewed, bimodal, etc., and understanding this helps in selecting the right statistical tools and correctly interpreting data pathways.

  • A normal shape is symmetrical with a single peak. It's the classic bell curve referenced often in statistics.
  • Skewed shapes, like those seen in exponential distributions, are asymmetrical, with tails extending either to the left or right.
  • Other shapes include bimodal (two peaks) and uniform (flat), each indicating different data patterns and necessitating different statistical analyses.
Grasping the shape of distribution aids in adapting analysis techniques to better model and understand data.

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

Glucose testing. Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood ghucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. In a test to screen for gestational diabetes, a patient is classified as needing further testing for gestational diabetes if the glucose level is above 130 milligrams per deciliter (mg/dL) one hour after having a sugary drink. Shelia's measured glucose level one hour after the sugary drink varies according to the Normal distribution with \(\mu=122 \mathrm{mg} / \mathrm{dL}\) and \(\sigma=12 \mathrm{mg} / \mathrm{dL}\). (a) If a single glucose measurement is made, what is the probability that Shelia is diagnosed as needing further testing for gestational diabetes? (b) If measurements are made on four separate days and the mean result is compared with the criterion \(130 \mathrm{mg} / \mathrm{dL}\), what is the probability that Shelia is diagnosed as needing further testing for gestational diabetes?

Playing the numbers: The house has a business. Unlike Joe (see the previous exercise), the operators of the numbers racket can rely on the law of large numbers. It is said that the New York Ciry mobster Casper Holstein took as many as 25,000 bets per day in the Prohibition era. That's 150,000 bets in a week if he takes Sunday off. Casper's mean winnings per bet are \(\$ 0.40\) (he pays out 60 cents of each dollar bet to people like Joe and keeps the other 40 cents). His standard deviation for single bets is about \(\$ 18.96\), the same as Joe's.New York Daily News 脗rchived Getty Images (a) What are the mean and standard deviation of Casper's average winnings \(x^{-} \bar{x}\) on his 150,000 bets? (b) According to the central limit theorem, what is the approximate probability that Casper's average winnings per bet are between \(\mathrm{S} 0.30\) and \(\$ 0.50 ?\) After only a week, Casper can be pretty confident that his winnings will be qu矛te close to \(\$ 0.40\) per bet.

Runners. In a study of exercise, a large group of male runners walk on a treadmill for six minutes. After this exercise, their heart rates vary with mean 8.8 beats per five seconds and standard deviation \(1.0\) beats per five seconds. This distribution takes only whole-number values, so it is certainly not Normal. (a) Let \(x^{-} \bar{x}\) be the mean number of beats per five seconds after measuring heart rate for 24 five-second intervals (two minutes). What is the approximate distribution of \(x^{-} x\) according to the central limit theorem? (b) What is the approximate probability that \(x^{-} x\) is less than 8 ? (c) What is the approximate probability that the heart rate of a runner is less than 100 beats per minute? (Hint: Restate this event in terms of \(x^{-}{ }^{3}\).)

Glucose testing, continued. Shelia's measured glucose level one hour after having a sugary drink varies according to the Normal distribution with \(\mu=122\) \(\mathrm{mg} / \mathrm{dL}\) and \(\sigma=12 \mathrm{mg} / \mathrm{dL}\). What is the level \(L\) such that there is probability only \(0.05\) that the mean glucose level of four test results falls above L? (Hint: This requires a backward Normal calculation. See page 91 in Chapter 3 if you need to review.)

Testing glass. How well materials conduct heat matters when designing houses. As a test of a new measurement process, 10 measurements are made on pieces of glass known to have conductivity 1 . The average of the 10 measurements is 1.07. For each of the boldface numbers, indicate whether it is a parameter or a statistic. Explain your answer.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.