10.4 Matched or Paired Samples
- Page ID
- 36523
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Section 10.4 Matched or Paired Samples
Learning Objective:
In this section, you will:
• Apply hypothesis testing and calculate confidence intervals to real-world problems about the mean of two dependent samples (matched pairs)
When using a hypothesis test for matched or paired samples, the following characteristics should be present:
- Simple random sampling is used.
- Sample sizes are often small.
- Two measurements (samples) are drawn from the same pair of individuals or objects.
- Differences are calculated from the matched or paired samples.
- The differences form the sample that is used for the hypothesis test.
- Either the matched pairs have differences that come from a population that is normal or the number of differences is sufficiently large so that distribution of the sample mean of differences is approximately normal.
A hypothesis test for matched or paired samples
- Random Variable: 𝑥̅𝑑 = mean of the differences
- Distribution: Student’s-t distribution with n – 1 degrees of freedom
Example 1: A study was conducted to investigate the effectiveness of hypnotism in reducing pain. Results for randomly selected subjects are shown in the table below. A lower score indicates less pain. The "before" value is matched to an "after" value and the differences are calculated. The differences have a normal distribution. Are the sensory measurements, on average, lower after hypnotism? Test at a 5% significance level.
|
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 |
First find the differences:
- Null and Alternative Hypothesis
1
- Calculator Work
- Test Statistic and P-Value
- Conclusion about the null hypothesis
- Final conclusion that addresses the original claim
- Test the above claim by constructing an appropriate confidence interval.
Example 2: 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 data were collected and recorded in the table below.
|
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.
- Null and Alternative Hypothesis
2
- Calculator Work
- Test Statistic and P-Value
- Conclusion about the null hypothesis
- Final conclusion that addresses the original claim
- Test the above claim by constructing an appropriate confidence interval.
For more information and examples see online textbook OpenStax Introductory Statistics pages 584-590.
“Introduction to Statistics” by OpenStax, used is licensed under a Creative Commons Attribution License 4.0 license