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Inside Collection (Textbook): Collaborative Statistics (MT230 - Spring 2013)
Summary: This module provides an overview of Hypothesis Testing: Matched or Paired Samples as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.
In a hypothesis test for matched or paired samples, subjects are matched in
pairs and differences are calculated. The differences are the data. The
population mean for the differences,
A study was conducted to investigate the effectiveness of hypnotism in reducing pain. Results for randomly selected subjects are shown in the table. The "before" value is matched to an "after" value and the differences are calculated. The differences have a normal distribution.
| Subject: | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Before | 6.6 | 6.5 | 9.0 | 10.3 | 11.3 | 8.1 | 6.3 | 11.6 |
| After | 6.8 | 2.4 | 7.4 | 8.5 | 8.1 | 6.1 | 3.4 | 2.0 |
Are the sensory measurements, on average, lower after hypnotism? Test at a 5% significance level.
Corresponding "before" and "after" values form matched pairs. (Calculate "sfter" - "before").
| After Data | Before Data | Difference |
|---|---|---|
| 6.8 | 6.6 | 0.2 |
| 2.4 | 6.5 | -4.1 |
| 7.4 | 9 | -1.6 |
| 8.5 | 10.3 | -1.8 |
| 8.1 | 11.3 | -3.2 |
| 6.1 | 8.1 | -2 |
| 3.4 | 6.3 | -2.9 |
| 2 | 11.6 | -9.6 |
The data for the test are the differences: {0.2, -4.1, -1.6, -1.8, -3.2, -2, -2.9, -9.6}
The sample mean and sample standard
deviation of the differences are:
Let
Random Variable:
Distribution for the test: The distribution is a student-t with
Calculate the p-value using the Student-t distribution:
Graph:
 |
The sample mean and sample standard deviation of the differences are:
Compare
Make a decision: Since
This means that
Conclusion: At a 5% level of significance, from the sample data, there is sufficient evidence to conclude that the sensory measurements, on average, are lower after hypnotism. Hypnotism appears to be effective in reducing pain.
STAT
and arrow over to TESTS. Press 2:T-Test. Arrow over to Data and press
ENTER. Arrow down and enter 0 for 1 for Freq:. Arrow down to μ: and arrow over to < ENTER. Arrow down to Calculate and press ENTER. The p-value is
0.0094 and the test statistic is -3.04. Do these instructions again except
arrow to Draw (instead of Calculate). Press ENTER.
A college football coach was interested in whether the college's strength development class increased his players' maximum lift (in pounds) on the bench press exercise. He asked 4 of his players to participate in a study. The amount of weight they could each lift was recorded before they took the strength development class. After completing the class, the amount of weight they could each lift was again measured. The data are as follows:
| Weight (in pounds) | Player 1 | Player 2 | Player 3 | Player 4 |
|---|---|---|---|---|
| Amount of weighted lifted prior to the class | 205 | 241 | 338 | 368 |
| Amount of weight lifted after the class | 295 | 252 | 330 | 360 |
The coach wants to know if the strength development class makes his players stronger, on average.
Record the differences data. Calculate the differences by subtracting the amount of weight lifted prior to the class from the weight lifted after completing the class. The data for the differences are: {90, 11, -8, -8}. The differences have a normal distribution.
Using the differences data, calculate the sample mean and the sample standard deviation.
Using the difference data, this becomes a test of a single __________ (fill in the blank).
Define the random variable:
The distribution for the hypothesis test is
Graph:
 |
Calculate the p-value: The p-value is 0.2150
Decision: If the level of significance is 5%, the decision is to not reject the null
hypothesis because
What is the conclusion?
means; At a 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the strength development class helped to make the players stronger, on average.
Seven eighth graders at Kennedy Middle School measured how far they could push the shot-put with their dominant (writing) hand and their weaker (non-writing) hand. They thought that they could push equal distances with either hand. The following data was collected.
| Distance (in feet) using | Student 1 | Student 2 | Student 3 | Student 4 | Student 5 | Student 6 | Student 7 |
|---|---|---|---|---|---|---|---|
| Dominant Hand | 30 | 26 | 34 | 17 | 19 | 26 | 20 |
| Weaker Hand | 28 | 14 | 27 | 18 | 17 | 26 | 16 |
Conduct a hypothesis test to determine whether the mean difference in distances between the children's dominant versus weaker hands is significant.
What is your conclusion?
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This work is licensed by Maxfield Foundation under a Creative Commons Attribution License (CC-BY 3.0), and is an Open Educational Resource.
Last edited by Susan Dean on Jun 15, 2012 11:40 am -0500.
"Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"