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Chapter 9: Problem 578

Briefly discuss the Central Limit Theorem.

Short Answer

Expert verified
The Central Limit Theorem (CLT) states that for a large enough sample size \(n\), the distribution of the sum or average of independent, identically distributed random variables with any distribution, mean \(渭\), and variance \(蟽^2\) approaches a normal distribution. This theorem is crucial in statistical inference, as it allows us to make predictions based on the normal distribution even if the underlying population distribution is not normal. For example, if we take a random sample of \(n = 50\) students' heights and calculate their average height multiple times, the distribution of these sample averages will be approximately normally distributed, allowing us to make statistical inferences about the population mean height based on our samples.

Step by step solution

01

Understand the Central Limit Theorem statement

The Central Limit Theorem states that if \(X_1, X_2, ..., X_n\) are independent, identically distributed random variables with any distribution (it doesn't have to be normal), mean \(渭\), and variance \(蟽^2\), then the distribution of their sum (\(S_n = X_1 + X_2 + ... + X_n\)) or average (\(\overline{X} = \frac{X_1 + X_2 + ... + X_n}{n}\)) approaches a normal distribution with mean \(n渭\) (for the sum) or \(渭\) (for the average), and variance \(n蟽^2\) (for the sum) or \(\frac{蟽^2}{n}\) (for the average) as \(n\) becomes large. In other words, for a large enough \(n\), the distribution of the sum or average is approximately normal.
02

Importance of the Central Limit Theorem

The Central Limit Theorem plays a fundamental role in statistical inference, as it allows us to make predictions (confidence intervals, hypothesis testing) based on the normal distribution, even if the underlying population distribution is not normal. The properties of the normal distribution, such as being symmetric and having the empirical rule (68-95-99.7 rule), make it convenient for working with probability and estimating population parameters. The approximation to normality improves as the sample size, \(n\), increases, which contributes to its broad applicability.
03

Example to illustrate the Central Limit Theorem

Let's say we have a population of students of various heights, and we are interested in the average height of students. The heights of individual students may not be normally distributed, but we can still use the Central Limit Theorem in this case. Suppose we take a random sample of \(n = 50\) students and compute their average height. According to the CLT, the distribution of the sample average height will approach a normal distribution as the sample size increases, even if the individual heights do not follow a normal distribution. So, if we repeat this process multiple times (e.g., 1,000 times) by taking different samples of 50 students each time and calculating their average height, then the distribution of the 1,000 sample averages will be approximately normally distributed. This approximation allows us to make statistical inferences about the population mean height based on our samples.

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