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Chapter 15: Problem 23

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, page \(\underline{355}\), 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
c. The average failure time of a large number of batteries has a distribution that is close to Normal.

Step by step solution

01

Understanding the Exponential Distribution

The problem mentions that the battery failure times follow an exponential distribution, which is strongly skewed to the right. This means that most batteries will fail relatively soon, but there are some that will last much longer, creating a long tail on the right side of the distribution.
02

Recalling the Central Limit Theorem (CLT)

The Central Limit Theorem states that as the size of a sample increases, the distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution, given that the original variable has a finite variance.
03

Applying the Central Limit Theorem

Given the CLT, when we take the average failure time of a large number of batteries, this average will start to form a normal distribution. This occurs even though the failure times for individual batteries follow an exponential distribution, because the CLT applies to the average of a sample, not the individual data.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Distribution
The exponential distribution is a type of statistical distribution that is often used to model the time until an event occurs, such as the lifespan of a product. In the context of our exercise, it refers to the failure times of batteries. A key characteristic of this distribution is that it is strongly skewed to the right. This means that there are many occurrences of short lifespans, but there can also be some instances of much longer lifespans. The skewness results in a long tail on the right side of the graph.

Some important properties of the exponential distribution include:
  • It is memoryless, meaning the probability of failure in the next moment is constant, regardless of how long it has already lasted.
  • The mean and the standard deviation are equal, which makes calculations quite convenient.
Sample Mean
The sample mean is a fundamental concept in statistics, representing the average of a set of data points. In our exercise, it's about calculating the average failure time of multiple batteries. This average is particularly useful when we want to understand the overall performance or behavior of a data set.

The formula to compute the sample mean is fairly simple:
  • Add up all the individual data points.
  • Divide the total by the number of data points.
As you increase your sample size, the sample mean becomes a more accurate estimate of the population mean. This ties into the Central Limit Theorem, which we'll discuss further.
Normal Distribution
The normal distribution, often referred to as the Gaussian distribution, is a common way to describe how data is spread out. It's shaped like a bell curve, where most values cluster around the mean, and the probabilities of extreme values are low. It's symmetric, meaning the left and right sides of the graph are mirror images.

This distribution is extensively important in statistics and is characterized by:
  • A mean of zero and a standard deviation of one in its standard form.
  • About 68% of data within one standard deviation of the mean, 95% within two, and 99.7% within three (68-95-99.7 rule).
The Central Limit Theorem suggests that the sample mean will form this normal distribution pattern if the sample size is large, even if the population data does not follow a normal distribution.
Statistical Distribution
A statistical distribution describes how the values of a variable are spread or arranged. Understanding distributions allows statisticians to make predictions and infer properties of a population from a sample. Distributions can be graphical representations like graphs or mathematical functions that define probabilities of different outcomes.

There are many types of statistical distributions, including:
  • Normal distribution
  • Exponential distribution
  • Binomial distribution
  • Poisson distribution
Knowing the type of distribution is crucial in selecting the right statistical methods for analysis. The original problem focused on the transition from exponential to normal distribution through sample means.
Variance
Variance is a measure of how much the data points in a data set differ from the mean and from each other. It's one of the key concepts in statistics to determine data variability. In simpler terms, it's about understanding how spread out a data set is.

Mathematically, variance is calculated as:
  • The average of the squared differences from the mean.
This involves:
  • Subtracting the mean from each data point, then squaring the result, which eliminates negative values.
  • Calculating the average of these squared differences.
A higher variance indicates that data points are more spread out. In our context, the Central Limit Theorem requires a finite variance of the underlying population distribution for the sample mean to approach a normal distribution.

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Most popular questions from this chapter

Detecting the Emerald Ash Borer. The emerald ash borer is a serious threat to ash trees. A state agriculture department places traps throughout the state to detect the emerald ash borer. When traps are checked periodically, the mean number of ash borers trapped is only \(2.2\), but some traps have many ash borers. The distribution of ash borer counts is finite and strongly skewed, with standard deviation \(3.9 .\) a. What are the mean and standard deviation of the average number of ash borers \(x\) in 50 traps? b. Use the central limit theorem to find the probability that the average number of ash borers in 50 traps is greater than 3.0.

Daily Activity. It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking. I Among mildly obese people, the mean number of minutes of daily activity (standing or walking) is approximately Normally distributed with mean 373 minutes and standard deviation 67 minutes. The mean number of minutes of daily activity for lean people is approximately Normally distributed with mean 526 minutes and standard deviation 107 minutes. A researcher records the minutes of activity for an SRS of five mildly obese people and an SRS of five lean people. a. What is the probability that the mean number of minutes of daily activity of the five mildly obese people exceeds 420 minutes? b. What is the probability that the mean number of minutes of daily activity of the five lean people exceeds 420 minutes?

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 89 in Chapter 3 if you need a review.)

Playing the Numbers: A Gambler Gets Chance Outcomes. The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers racket described in the previous exercise has outcomes that vary considerably-one three-digit number wins \(\$ 600\) and all others win nothing - that gamblers never reach "the long run." Even after many bets, their average winnings may not be close to the mean. For the numbers racket, the mean payout for single bets is \(\$ 0.60\) (60 cents), and the standard deviation of payouts is about \(\$ 18.96\). If Joe plays 350 days a year for 40 years, he makes 14,000 bets. a. What are the mean and standard deviation of the average payout \(x\) that Joe receives from his 14,000 bets? b. The central limit theorem says that his average payout is approximately Normal with the mean and standard deviation you found in part (a). What is the approximate probability that Joe's average payout per bet is between \(\$ 0.50\) and \(\$ 0.70\) ? You see that Joe's average may not be very close to the mean \(\$ 0.60\) even after 14,000 bets.

Annual returns on stocks vary a lot. The long-term mean return on stocks in the S\&P 500 is \(9.8 \%\), and the long-term standard deviation of returns is \(16.8 \%\). The law of large numbers says that a. you can get an average return higher than the mean \(9.8 \%\) by investing in a large number of the \(\mathrm{S} \& \mathrm{P}\) stocks. b. as you invest in more and more stocks chosen at random, your long-term average return on these stocks gets ever closer to \(9.8 \%\). c. if you invest in a large number of stocks chosen at random, your long-term average return will have approximately a Normal distribution.
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