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Chapter 3: Problem 47

Suppose we have \(n\) independent observations \(X_{1}, X_{2}, \ldots, X_{n}\) from a distribution with mean \(\mu\) and standard deviation \(\sigma .\) What is the variance of the mean of these \(n\) values: \(\frac{X_{1}+X_{2}+\cdots+X_{n}}{n} ?\)

Short Answer

Expert verified
The variance of the mean is \(\frac{\sigma^2}{n}\).

Step by step solution

01

Identify the Given Variables

We have \(n\) independent observations \(X_{1}, X_{2}, \ldots, X_{n}\) from a distribution with a mean of \(\mu\) and standard deviation \(\sigma\). We need to find the variance of their mean, which is expressed as \(\bar{X} = \frac{X_{1}+X_{2}+\cdots+X_{n}}{n}\).
02

Apply the Variance Formula for a Sum

The variance of a sum of independent random variables is the sum of their variances. Therefore, the variance of \(S = X_1 + X_2 + \cdots + X_n\) is \(\text{Var}(S) = n\sigma^2\) because each \(X_i\) has variance \(\sigma^2\) and they are independent.
03

Calculate the Variance of the Mean

The variance of the average, \(\bar{X} = \frac{S}{n}\), is given by \(\text{Var}(\bar{X}) = \text{Var}\left(\frac{S}{n}\right) = \frac{1}{n^2} \times \text{Var}(S)\). Substitute \(\text{Var}(S) = n\sigma^2\) to get \(\text{Var}(\bar{X}) = \frac{1}{n^2} \times n\sigma^2 = \frac{\sigma^2}{n}\).

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Key Concepts

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

Independent Observations
When we talk about independent observations, we're referring to random variables that do not affect each other. This means that the occurrence of one observation does not change the probability of occurrence of another. Imagine flipping a fair coin multiple times. Each flip is independent of the others. Whether it lands heads or tails on one flip doesn't impact the outcome of the next flip.
  • Independence: Observations don't influence each other.
  • Applications: Used in statistical models to simplify analysis.
  • Significance: Ensures accurate estimations and calculations.
Independence is crucial when calculating the variance because it allows us to add the variances of individual observations without worrying about interactions.
Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are. A smaller standard deviation means the numbers are closer to the average, while a larger one indicates more spread out data. This statistic gives us an understanding of how much the observations vary from the mean.
For example, if you have a distribution of students' heights, the standard deviation tells you how much individual heights differ from the average height of the group.
  • Definition: A measure of dispersion or spread in a dataset.
  • Symbol: Denoted as \( \sigma \) in equations.
  • Importance: Helps determine the variability within a dataset.
In the context of our problem, it's important because it helps us calculate the variance of the sample mean.
Mean of a Distribution
The mean, often referred to as the average, is a central value representing a set of numbers. In a distribution, the mean is calculated by summing all values and dividing by the number of values.
It gives a quick sense of the overall trend of a dataset. For instance, if we are looking at exam scores, the mean score can tell us how the class performed as a whole.
  • Formula: \( \mu = \frac{\sum X}{n} \), where \( X \) are observations.
  • Significance: Represents central tendency or the average.
  • Role: A foundational concept in statistics used for further analysis.
In our problem, knowing the mean allows us to explore how individual observations deviate from this central value, impacting calculations like variance.
Random Variables
A random variable is a variable whose possible values are outcomes of a random phenomenon. Unlike a traditional variable, it is associated with random processes and can take on different values.
For instance, rolling a die results in a random variable that can have values between 1 and 6. Each possible value has a probability associated with it.
  • Definition: An outcome variable determined by chance.
  • Types: Can be discrete (specific values) or continuous (range of values).
  • Purpose: Used to model uncertainty and randomness in data.
In the exercise, each \( X_i \) is a random variable. Understanding these helps us to calculate overall properties like the mean and variance.

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Most popular questions from this chapter

The Behavioral Risk Factor Surveillance System (BRFSS) is an annual telephone survey designed to identify risk factors in the adult population and report emerging health trends. The following table summarizes two variables for the respondents: health status and health coverage, which describes whether each respondent had health insurance. \({ }^{64}\) $$\begin{array}{rrrrrrr} & {\text { Health Status }} && \text { Excellent } & \text { Very good } & \text { Good } & \text { Fair } & \text { Poor } & \text { Total } \\\\\hline \text { No } & 459 & 727 & 854 & 385 & 99 & 2,524 \\ \text { Yes } & 4,198 & 6,245 & 4,821 & 1,634 & 578 & 17,476 \\ \hline \text { Total } & 4,657 & 6,972 & 5,675 & 2,019 & 677 & 20,000 (a) If we draw one individual at random, what is the probability that the respondent has excellent health and doesn't have health coverage? (b) If we draw one individual at random, what is the probability that the respondent has excellent health or doesn't have health coverage? \end{array}$$

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Guessing on an exam. In a multiple choice exam, there are 5 questions and 4 choices for each question \((\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}) .\) Nancy has not studied for the exam at all and decides to randomly guess the answers. What is the probability that: (a) the first question she gets right is the \(5^{t h}\) question? (b) she gets all of the questions right? (c) she gets at least one question right?

An airline charges the following baggage fees: $$\$ 25$$ for the first bag and $$\$ 35$$ for the second. Suppose \(54 \%\) of passengers have no checked luggage, \(34 \%\) have one piece of checked luggage and \(12 \%\) have two pieces. We suppose a negligible portion of people check more than two bags. (a) Build a probability model, compute the average revenue per passenger, and compute the corresponding standard deviation. (b) About how much revenue should the airline expect for a flight of 120 passengers? With what standard deviation? Note any assumptions you make and if you think they are justified.

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