Question

Instead of finding the mean of the differences between $X_{1}$ and $X_{2}$ by subtracting $X_{1}-X_{2},$ you can find it by finding the means of $X_{1}$ and $X_{2}$ and then subtracting the means. Show that these two procedures will yield the same results.

Name: Problem 13, Chapter 9: Elementary Statistics a Step by Step Approach
Uploaded: 2020-07-29T20:49:24-08:00
Duration: 3 min 38 s
Channel: Robin Corrigan
Description: Instead of finding the mean of the differences between X1 and X2 by subtracting X1-X2, you can find it by finding the means of X1 and X2 and then subtracting the means. Show that these two procedures will yield the same results.

   Instead of finding the mean of the differences between $X_{1}$ and $X_{2}$ by subtracting $X_{1}-X_{2},$ you can find it by finding the means of $X_{1}$ and $X_{2}$ and then subtracting the means. Show that these two procedures will yield the same results.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Chapter 9, Problem 13

Instant Answer

Solved by Expert Robin Corrigan

07/29/2020

Step 1

The first procedure is to find the mean of the differences between $X_{1}$ and $X_{2}$ by subtracting $X_{1}-X_{2}$ for each pair of samples and then averaging these differences. This can be represented as $\overline{X_{1}-X_{2}}$. Show more…

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Instead of finding the mean of the differences between $X_{1}$ and $X_{2}$ by subtracting $X_{1}-X_{2},$ you can find it by finding the means of $X_{1}$ and $X_{2}$ and then subtracting the means. Show that these two procedures will yield the same results.

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Key Concepts

Random Variables and Expected Value

Random variables are quantities that can vary and their expected value, or mean, is the long-run average of their outcomes. The expected value serves as a measure of the central tendency of a random variable and provides a single summary value that represents the average outcome.

Linearity of Expectation

The linearity of expectation is a fundamental property in probability theory stating that the expected value of the sum or difference of random variables is equal to the sum or difference of their individual expected values. This implies that when calculating the mean difference between two random variables, one can either compute the difference for each pair and then average or subtract the averages of the individual variables to obtain the same result.

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