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Chapter 3: Problem 46

Suppose we have 3 independent observations \(X_{1}, X_{2}, X_{3}\) from a distribution with mean \(\mu\) and standard deviation \(\sigma .\) What is the variance of the mean of these 3 values: \(\frac{X_{1}+X_{2}+X_{3}}{3} ?\)

Short Answer

Expert verified
The variance of the mean is \(\frac{\sigma^2}{3}\).

Step by step solution

01

Understand the Concept of Mean

To find the variance of the mean of three observations, \(\bar{X} = \frac{X_1 + X_2 + X_3}{3}\), we need to understand how the variance of a sample mean works. The key concept is that the variance of the sample mean from \(n\) independent observations is the variance of one observation divided by \(n\).
02

Identify the Variance Formula

Since \(X_1, X_2, X_3\) are independent and identically distributed with variance \(\sigma^2\), each individual observation has variance \(\text{Var}(X_i) = \sigma^2\). For \(3\) independent observations, the variance of their mean is calculated as follows.
03

Apply the Variance Formula to the Mean

The variance of the mean \(\bar{X}\) is given by \(\text{Var}(\bar{X}) = \frac{\sigma^2}{n}\), where \(n\) is the number of observations. Substituting \(n = 3\) and \(\sigma^2\) being the variance of each observation gives us: \[\text{Var}(\bar{X}) = \frac{\sigma^2}{3}.\]
04

State the Result

The variance of the mean of the three observations \(X_1, X_2, X_3\) is thus \(\frac{\sigma^2}{3}\). This tells us how much the mean of our observations is expected to deviate from \(\mu\) when averaged.

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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 mean that each observation in a sample does not influence or affect any other observation. Imagine you're measuring the heights of apples on different trees. Each tree's height measurement is independent of the others because each tree grows in its own way.
  • This independence is crucial for many statistical methods, including calculating the variance of a sample mean.
  • If the observations were not independent, finding true variance would be more complicated, as you'd need to account for how they influence each other.
  • In our original problem, the observations \(X_1, X_2, X_3\) are independent, meaning each has its own independent variability.
Understanding independence helps simplify the process of calculating variance because you can think of each observation in isolation, without needing to worry about covariance between them.
Identically Distributed
Identically distributed observations imply that each observation in the sample comes from the same probability distribution. They share the same mean \(\mu\) and the same variance \(\sigma^2\).
  • If you consider drawing marbles from a bag, identically distributed means each draw has the same likelihood of a specific outcome 鈥� like each marble being equally likely to be blue or red if the bag's contents don't change.
  • In our problem, \(X_1, X_2, X_3\) all come from the same distribution, thus they are identically distributed.
  • This uniformity allows us to use the same statistical properties (mean and variance) for all observations, simplifying calculations and analyses.
By having identically distributed observations, we can easily apply statistical theories, like finding the variance of the sample mean, across all observations.
Mean of Observations
The mean of observations refers to the process of averaging a set of numbers to find their central value. For our example with \(X_1, X_2, X_3\), the mean \(\bar{X}\) is calculated as \(\frac{X_1 + X_2 + X_3}{3}\).
  • The mean provides a simple summary of the dataset and often serves as a best guess value for predicting future outcomes from the same distribution.
  • When dealing with several observations, calculating their mean helps to reduce the effect of random fluctuations or outliers in individual observations.
  • In the context of variance calculation, knowing the mean helps understand how typical values deviate from expected outcomes.
The mean is a foundational concept in statistics and gives insights into the average behavior of data points within a dataset.
Sample Variance
Sample variance is a measure of how much the observations in a sample deviate from their mean. It's calculated by averaging the squared differences between each observation and the mean of the sample.
  • For independent and identically distributed observations, the sample variance is representative of the underlying distribution's variance.
  • The formula for the variance of a single observation is \(\sigma^2\), but for a sample of \(n\) observations, the variance of the mean becomes \(\frac{\sigma^2}{n}\).
  • This formula shows that as the sample size increases, the variance of the mean decreases, meaning that larger samples give more reliable estimates.
Understanding how to calculate and interpret sample variance is crucial for assessing the reliability and variability of statistical estimates based on sample data.

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Most popular questions from this chapter

Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 204 Scandinavian men and their female partners. The table below summarizes the results. For simplicity, we only include heterosexual relationships in this exercise. \(^{42}\) $$\begin{array}{lcccc} & {\text { Partner (female) }} & \\ & \text { Blue } & \text { Brown } & \text { Green } & \text { Total } \\ \hline \text { Blue } & 78 & 23 & 13 & 114 \\ \text { Brown } & 19 & 23 & 12 & 54 \\ \text { Green } & 11 & 9 & 16 & 36 \\ \hline \text { Total } & 108 & 55 & 41 & 204\end{array}$$ (a) What is the probability that a randomly chosen male respondent or his partner has blue eyes? (b) What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes? (c) What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes? What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes? (d) Does it appear that the eye colors of male respondents and their partners are independent? Explain your reasoning.

If you roll a pair of fair dice, what is the probability of (a) getting a sum of \(1 ?\) (b) getting a sum of \(5 ?\) (c) getting a sum of \(12 ?\)

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?

Swaziland has the highest HIV prevalence in the world: \(25.9 \%\) of this country's population is infected with HIV. \(^{65}\) The ELISA test is one of the first and most accurate tests for HIV. For those who carry HIV, the ELISA test is \(99.7 \%\) accurate. For those who do not carry HIV, the test is \(92.6 \%\) accurate. If an individual from Swaziland has tested positive, what is the probability that he carries HIV?

A 2010 SurveyUSA poll asked 500 Los Angeles residents, "What is the best hamburger place in Southern California? Five Guys Burgers? In-N-Out Burger? Fat Burger? Tommy's Hamburgers? Umami Burger? Or somewhere else?" The distribution of responses by gender is shown below. \({ }^{41}\) $$\begin{array}{lrrr} & {\text { Gender }} & \\ & \text { Male } & \text { Female } & \text { Total } \\\\\hline \text { Five Guys Burgers } & 5 & 6 & 11 \\ \text { In-N-Out Burger } & 162 & 181 & 343 \\\\\text { Fat Burger } & 10 & 12 & 22 \\ \text { Tommy's Hamburgers } & 27 & 27 & 54 \\\\\text { Umami Burger } & 5 & 1 & 6 \\ \text { Other } & 26 & 20 & 46 \\\\\text { Not Sure } & 13 & 5 & 18 \\ \hline \text { Total } & 248 & 252 & 500\end{array}$$ (a) Are being female and liking Five Guys Burgers mutually exclusive? (b) What is the probability that a randomly chosen male likes In-N-Out the best? (c) What is the probability that a randomly chosen female likes In-N-Out the best? (d) What is the probability that a man and a woman who are dating both like In-N-Out the best? Note any assumption you make and evaluate whether you think that assumption is reasonable. (e) What is the probability that a randomly chosen person likes Umami best or that person is female?
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