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Chapter 8: Problem 2

State the Central Limit Theorem.

Short Answer

Expert verified
The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the population's distribution.

Step by step solution

01

Understand the Context

The Central Limit Theorem (CLT) is a fundamental result in probability theory and statistics. It applies to the distribution of sample means.
02

Sample Selection

Consider a large number of independent random samples, each of size n, taken from a population with mean \(\mu\) and standard deviation \(\sigma\).
03

Distribution of Sample Means

When the sample size n is sufficiently large, the distribution of the sample means \(\bar{X}\) will approach a normal distribution, regardless of the original population distribution.
04

Mathematical Expression

The Central Limit Theorem can be mathematically expressed as: If X鈧�, X鈧�, ..., X鈧� are independent random variables with mean \(\mu\) and variance \(\sigma^2\), then the distribution of the sample mean \(\bar{X}\) is approximately normal with mean \(\mu\) and standard deviation \(\frac{\sigma}{\sqrt{n}}\). Formally, \[\bar{X} \sim N \(\mu, \frac{\sigma}{\sqrt{n}}\)\].
05

Conditions for CLT

The theorem works under the condition that the sample size is large (often n > 30 is considered sufficient), and the samples are independent.

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Key Concepts

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

Sample Means Distribution
When we take multiple samples from a population and calculate the mean of each sample, we get a set of means.
This set of means is called the *distribution of sample means*.
According to the Central Limit Theorem (CLT), this distribution tends to follow a normal distribution as the sample size increases.
This fascinating result happens regardless of the population鈥檚 original distribution.
So whether the original data is skewed or heavily tailed, the sample means will form a bell-shaped curve as we draw more samples.
The sample means distribution is crucial because it allows us to make inferences and predictions about the population with less variability.
Normal Distribution
A normal distribution, also known as the bell curve, is a fundamental concept in statistics.
It's symmetrical, meaning the left and right sides of the curve are mirror images.
Most data points cluster around the central peak, with fewer points further away.
This pattern represents many natural phenomena, like heights or test scores.
In the context of the Central Limit Theorem, the sample means distribution will approximate a normal distribution under sufficient sample size.
This approximation helps simplify many statistical calculations and makes the concepts easier to handle.
Sampling Conditions
To apply the Central Limit Theorem effectively, certain conditions need to be fulfilled:
  • The sample size should be large, typically more than 30.
  • The samples must be independent.
If these conditions are met, the distribution of the sample means will approximate a normal distribution.
If the sample size is too small, the approximation may not be valid, leading to inaccurate inferences.
These conditions ensure that the theorem holds true and the predictions we make are reliable.
Probability Theory
Probability theory is a branch of mathematics concerned with analyzing random events. The Central Limit Theorem is a critical result in this theory.
The theorem relies on the idea that as we increase the number of trials (samples), the probability distribution of the sample means converges to a normal distribution.
This convergence allows us to use tools and concepts from probability theory to analyze and predict outcomes.
Understanding probability theory is essential for grasping why the Central Limit Theorem works and its implications for statistics and real-world data analysis.
Independent Random Samples
Independence of samples is key for the Central Limit Theorem to work correctly.
Two samples are independent if the selection of one sample does not affect the selection of the other.
For example, when choosing students randomly from different schools to measure heights, each choice should be unrelated to the others.
This independence ensures that the samples provide a true representation of the population and that the properties displayed by the sample means distribution are valid.
Without independence, results could be biased or misleading, undermining the reliability of statistical analyses.

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Most popular questions from this chapter

The Food and Drug Administration sets Food Defect Action Levels (FDALs) for some of the various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL for insect filth in peanut butter is 3 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A random sample of 50 ten-gram portions of peanut butter is obtained and results in a sample mean of \(\bar{x}=3.6\) insect fragments per ten-gram portion. (a) Why is the sampling distribution of \(\bar{x}\) approximately normal? (b) What is the mean and standard deviation of the sampling distribution of \(\bar{x}\) assuming \(\mu=3\) and \(\sigma=\sqrt{3}\) (c) Suppose a simple random sample of \(n=50\) ten-gram samples of peanut butter results in a sample mean of 3.6 insect fragments. What is the probability a simple random sample of 50 ten-gram portions results in a mean of at least 3.6 insect fragments? Is this result unusual? What might we conclude?

Based on tests of the Chevrolet Cobalt, engineers have found that the miles per gallon in highway driving are normally distributed, with a mean of 32 miles per gallon and a standard deviation 3.5 miles per gallon. (a) What is the probability that a randomly selected Cobalt gets more than 34 miles per gallon? (b) Suppose that 10 Cobalts are randomly selected and the miles per gallon for each car are recorded. What is the probability that the mean miles per gallon exceed 34 miles per gallon? (c) Suppose that 20 Cobalts are randomly selected and the miles per gallon for each car are recorded. What is the probability that the mean miles per gallon exceed 34 miles per gallon? Would this result be unusual?

In a 2003 study, the Accreditation Council for Graduate Medical Education found that medical residents work an average of 81.7 hours per week. Suppose the number of hours worked per week by medical residents is normally distributed with standard deviation 6.9 hours per week. (Source: www.medrecinst.com) (a) What is the probability that a randomly selected medical resident works less than 75 hours per week? (b) What is the probability that the mean number of hours worked per week by a random sample of five medical residents is less than 75 hours? (c) What is the probability that the mean number of hours worked per week by a random sample of eight medical resident is less than 75 hours? (d) What might you conclude if the mean number of hours worked per week by a random sample of eight medical residents is less than 75 hours?

Explain what a sampling distribution is.

Gardeners Suppose a simple random sample of size \(n=100\) households is obtained from a town with 5000 households. It is known that \(30 \%\) of the households plant a garden in the spring. (a) Describe the sampling distribution of \(\hat{p}\) (b) What is the probability that more than 37 households in the sample plant a garden? Is this result unusual? (c) What is the probability that 18 or fewer households in the sample plant a garden? Is this result unusual?
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