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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 produce the same result: \(\frac{1}{n} \left( \sum_{i=1}^{n} X_{1i} - \sum_{i=1}^{n} X_{2i} \right)\).

Step by step solution

01

Understanding the Problem

We need to find the mean of the differences between two sets of numbers, \(X_1\) and \(X_2\), using two different methods: directly subtracting \(X_1-X_2\) and averaging, and by subtracting the means of \(X_1\) and \(X_2\). We will show that both methods yield the same result.
02

Calculating Direct Mean of Differences

First, calculate the differences for each pair of observations: \(d_i = X_{1i} - X_{2i}\). The mean of these differences is given by \(\bar{d} = \frac{1}{n} \sum_{i=1}^{n} (X_{1i} - X_{2i})\), which simplifies to \(\bar{d} = \frac{1}{n} \left( \sum_{i=1}^{n} X_{1i} - \sum_{i=1}^{n} X_{2i} \right)\).
03

Calculating Means and Subtracting

Calculate the mean of \(X_1\) using \(\bar{X_1} = \frac{1}{n}\sum_{i=1}^n X_{1i}\) and the mean of \(X_2\) using \(\bar{X_2} = \frac{1}{n}\sum_{i=1}^n X_{2i}\). Subtract these means: \(\bar{X_1} - \bar{X_2} = \frac{1}{n}\sum_{i=1}^n X_{1i} - \frac{1}{n}\sum_{i=1}^n X_{2i}\).
04

Comparing Results

Both methods yield the expression \(\frac{1}{n} \left( \sum_{i=1}^{n} X_{1i} - \sum_{i=1}^{n} X_{2i} \right)\). This proves that taking the mean of the differences is the same as subtracting the means.

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Key Concepts

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

Difference of Means
The concept of the "Difference of Means" is an essential part of understanding how sets of data compare. Imagine you have two sets of numbers, such as test scores from two different classes. To determine whose average performance is higher, you might look at the means of these sets. But another approach is to look at the differences in scores between the two classes paired one by one.
When we talk about the difference of means, we refer to subtracting the mean value of one set from the mean value of the other set. This calculation helps to understand overall differences between groups. For instance, if class A has an average score of 80 and class B has an average score of 75, the difference of means is 80 - 75 = 5. This indicates class A, on average, scored 5 points higher than class B.
Calculating the difference of means gives a quick insight into which of the two groups performs better on average.
Statistical Comparison
"Statistical Comparison" involves comparing different groups of data to uncover meaningful differences. It is a foundation for making informed decisions based on data analysis. This could involve comparing means, variances, or other statistical measures between groups.
  • Purpose: Comparing means allows us to see if there is a statistically significant difference between the groups. This might indicate a meaningful difference in real-world terms.
  • Method: Statistical tests such as the t-test can help determine if differences observed are due to random chance or a true distinction between groups.
Through statistical comparisons, like comparing the means of two different datasets as shown in the exercise, we gain insights into the data that help us draw conclusions or make predictions based on evidence.
Average of Differences
The "Average of Differences" is another way of comparing two datasets. Instead of directly calculating the difference between two means, you calculate the difference for each pair of corresponding values and then find the average of these differences.
To illustrate this, consider datasets representing heights of plants grown under different conditions. By computing the individual differences (the difference of each plant's height in condition A minus its counterpart in condition B) and then averaging these differences, you obtain the average of differences. This value gives a precise estimation of how each set varies relative to each other.
This method can offer a more detailed view into variations within paired data, sometimes offering insights that direct means comparison cannot offer. The key takeaway is understanding that both approaches, when applied correctly, should coincide in their conclusion.

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Most popular questions from this chapter

A researcher claims that students in a private school have exam scores that are at most 8 points higher than those of students in public schools. Random samples of 60 students from each type of school are selected and given an exam. The results are shown. At \(\alpha=0.05,\) test the claim. $$ \begin{array}{cc} \text { Private school } & \text { Public school } \\ \hline \bar{X}_{1}=110 & \bar{X}_{2}=104 \\ \sigma_{1}=15 & \sigma_{2}=15 \\ n_{1}=60 & n_{2}=60 \end{array} $$

Perform these steps. 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. National statistics show that \(23 \%\) of men smoke and \(18.5 \%\) of women smoke. A random sample of 180 men indicated that 50 were smokers, and a random sample of 150 women surveyed indicated that 39 smoked. Construct a \(98 \%\) confidence interval for the true difference in proportions of male and female smokers. Comment on your interval-does it support the claim that there is a difference?

Perform each of the following steps. 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. The average monthly Social Security benefit for a specific year for retired workers was \(\$ 954.90\) and for disabled workers was \(\$ 894.10 .\) Researchers used data from the Social Security records to test the claim that the difference in monthly benefits between the two groups was greater than \(\$ 30 .\) Based on the following information, can the researchers' claim be supported at the 0.05 level of significance? $$ \begin{array}{lll} & \text { Retired } & \text { Disabled } \\ \hline \text { Sample size } & 60 & 60 \\ \text { Mean benefit } & \$ 960.50 & \$ 902.89 \\ \text { Population standard deviation } & \$ 98 & \$ 101 \end{array} $$

The mean travel time to work for Americans is 25.3 minutes. An employment agency wanted to test the mean commuting times for college graduates and those with only some college. Thirty-five college graduates spent a mean time of 40.5 minutes commuting to work with a population variance of \(67.24 .\) Thirty workers who had completed some college had a mean commuting time of 34.8 minutes with a population variance of \(39.69 .\) At the 0.05 level of significance, can a difference in means be concluded?

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. Reducing Errors in Spelling A ninth-grade teacher wishes to see if a new spelling program will reduce the spelling errors in his students' writing. The number of spelling errors made by the students in a five-page report before the program is shown. Then the number of spelling errors made by students in a five-page report after the program is shown. At \(\alpha=0.05,\) did the program work? $$ \begin{array}{lllrllll} \text { Before } & 8 & 3 & 10 & 5 & 9 & 11 & 12 \\ \hline \text { After } & 6 & 4 & 8 & 1 & 4 & 7 & 11 \end{array} $$
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