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Chapter 9: Problem 13

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.

Short Answer

Expert verified
Both methods yield the same mean difference.

Step by step solution

01

Define Variables and Initial Setup

Let \( n \) be the number of data points. We have two datasets comprising \( X_1 = x_{11}, x_{12}, ..., x_{1n} \) and \( X_2 = x_{21}, x_{22}, ..., x_{2n} \). We are comparing two methods to find the mean of the differences between \( X_1 \) and \( X_2 \).
02

Procedure 1: Mean of Differences Directly

The mean of the differences when directly computed is: \[ \bar{d} = \frac{1}{n} \sum_{i=1}^{n} (x_{1i} - x_{2i}) = \frac{1}{n} (x_{11} - x_{21} + x_{12} - x_{22} + ... + x_{1n} - x_{2n}) \].
03

Procedure 2: Subtract Means of Each Set

First compute the mean of \( X_1 \): \( \bar{x}_1 = \frac{1}{n} \sum_{i=1}^{n} x_{1i} \). Then compute the mean of \( X_2 \): \( \bar{x}_2 = \frac{1}{n} \sum_{i=1}^{n} x_{2i} \). The difference of means is: \[ \bar{x}_1 - \bar{x}_2 = \frac{1}{n} \sum_{i=1}^{n} x_{1i} - \frac{1}{n} \sum_{i=1}^{n} x_{2i} = \frac{1}{n} \sum_{i=1}^{n} (x_{1i} - x_{2i}) \].
04

Comparison and Conclusion

Notice that both expressions for \( \bar{d} \) and \( \bar{x}_1 - \bar{x}_2 \) are the same: \( \frac{1}{n} \sum_{i=1}^{n} (x_{1i} - x_{2i}) \). This shows that both methods yield the same result.

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Key Concepts

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

Understanding Mean Difference
Mean difference is an important concept in statistics that allows us to compare two sets of data. It represents the average difference between corresponding values in two data sets.
There are two primary ways to calculate this mean difference:
  • Directly calculate the difference for each pair, then find the mean of these differences.
  • Find the mean of each set first and then subtract one mean from the other.
Using either method will yield the same result because of the properties of arithmetic mean. This equivalence holds due to the distributive nature of summation over subtraction.
Defining Population in Statistics
In the context of statistics, a population refers to the complete set of items or individuals that we are interested in studying or analyzing. A population can be very large, such as all the people living in a country, or smaller, like the attendees of a particular event.
When we talk about calculating statistics like mean difference, we often have a specific population in mind for each data set. It's important to distinguish when we're dealing with a whole population versus a sample, as this impacts the methods and conclusions in our analysis.
Working with Data Sets
A data set is simply a collection of related values or observations. In statistics, we often work with two data sets when calculating mean differences to make comparative analyses.
Each data set typically corresponds to a variable, and each value within these sets represents an observation. It's crucial to ensure data sets are properly aligned and well-defined to obtain meaningful results from statistical calculations.
For instance, in the exercise, data sets \(X_1\) and \(X_2\) are used to demonstrate how we can compute mean differences either directly or via mean subtraction.
Exploring Statistical Methods for Analysis
Statistical methods are techniques we use to analyze and make sense of data. They allow us to test hypotheses, draw conclusions, and make predictions.
Mean difference is just one of many statistical methods available. When choosing a method, it's essential to consider the nature of your data and the research question you are addressing.
In analyzing data sets for mean difference, statistical methods ensure precision and clarity in determining how two sets compare. These methods can guide us in selecting the appropriate formula or computational approach to investigate relationships within data.

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Most popular questions from this chapter

For Exercises 9 through \(24,\) perform the following steps. Assume that all variables are normally distributed. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. School Teachers' Salaries A researcher claims that the variation in the salaries of elementary school teachers is greater than the variation in the salaries of secondary school teachers. A random sample of the salaries of 30 elementary school teachers has a variance of \(8324,\) and a random sample of the salaries of 30 secondary school teachers has a variance of \(2862 .\) At \(\alpha=0.05\) can the researcher conclude that the variation in the elementary school teachers' salaries is greater than the variation in the secondary school teachers' salaries? Use the \(P\) -value method.

Find \(\hat{p}\) and \(\hat{q}\) for each. a. \(n=36, X=20\) b. \(n=50, X=35\) c. \(n=64, X=16\) d. \(n=200, X=175\) e. \(n=148, X=16\)

For Exercises 2 through \(12,\) perform each of these steps. Assume that all variables are normally or approximately normally distributed. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Pulse Rates of Identical Twins A researcher wanted to compare the pulse rates of identical twins to see whether there was any difference. Eight sets of twins were randomly selected. The rates are given in the table as number of beats per minute. At \(\alpha=0.01,\) is there a significant difference in the average pulse rates of twins? Use the \(P\) -value method. Find the \(99 \%\) confidence interval for the difference of the two. $$ \begin{array}{l|cccccccc}{\text { Twin } \mathbf{A}} & {87} & {92} & {78} & {83} & {88} & {90} & {84} & {93} \\ \hline \text { Twin B } & {83} & {95} & {79} & {83} & {86} & {93} & {80} & {86}\end{array} $$

Self-Esteem Scores In a study of a group of women science majors who remained in their profession and a group who left their profession within a few months of graduation, the researchers collected the data shown here on a self-esteem questionnaire. At \(\alpha=0.05,\) can it be concluded that there is a difference in the self-esteem scores of the two groups? Use the \(P\) -value method. $$ \begin{array}{ll}{\text { Leavers }} & {\text { Stayers }} \\\ {\bar{X}_{1}=3.05} & {\bar{X}_{2}=2.96} \\ {\sigma_{1}=0.75} & {\sigma_{2}=0.75} \\ {n_{1}=103} & {n_{2}=225}\end{array} $$

Miniature Golf Scores A large group of friends went miniature golfing together at a par 54 course and decided to play on two teams. A random sample of scores from each of the two teams is shown. At \(\alpha=0.05\) is there a difference in mean scores between the two teams? Use the \(P\) -value method. $$ \begin{array}{l|lllllll}{\text { Team } 1} & {61} & {44} & {52} & {47} & {56} & {63} & {62} & {55} \\ \hline \text { Team 2} & {56} & {40} & {42} & {58} & {48} & {52} & {51}\end{array} $$
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