Confidence Intervals: Matched Pairs 


Introduction

Matched Pairs confidence intervals are used when...

  • You take one random sample to study two paired variables.
    • These paired variables are DEPENDENT.
    • Usually, you are studying some measurable quality about the same subject before and after a treatment.
  • Your goal is to find the mean difference (\(\bar{x}_d\)).

The variables are said to be "paired" because you are comparing the two and because they are dependent. This last point is critical, because it allows you to tell apart a Matched Pairs confidence interval (which has dependent variables) and a 2 Sample Mean confidence interval (which has independent variables).

Let's look at an example of paired, dependent variables: you take a sample and measure each participant twice under two different experimental conditions.

  • We will call this data paired because we are interested in comparing a measurement from the participant under the first experimental condition to a measurement from the participant under the second experimental condition.
  • Furthermore, we can call the variables dependent because the measurements from experimental condition #1 and the measurements from experimental condition #2 are related to each other in the sense that both measurements were taken from the same participant. The numbers each participant yields might be very different, hence why we pair up each participant's measurements and then study the mean difference.

 

Example

You are interested in whether people prefer coke or pepsi. You take a sample of 36 individuals and set up a taste test. After each subject has tried a drink, he or she reports a score from 1 to 10. The data is shown below. Construct an 80% confidence interval for the mean difference in taste. 

Subject Coke Score Pepsi Score Difference
1 5 7 -2
2 2 1 +1
3 10 9 +1
... 36 9 5 +4

** Abbreviated table. Assume  \(\bar{x}_d = 0.682\) and \(s_d = 1.84\).

 

Step 1: Name the Confidence Interval: Matched Pairs

Step 2: Check the Conditions

1. The data is from a random sample.

2.  N ≥ 10n

3.  The sampling distribution of \(\bar{x}_d\) is approximately normal.

 
Step 3: Construct the Interval (Apply the Formula)
 
Matched Pairs Confidence Interval Formula: \(\bar{x}_d \pm t^{*}\frac{s_d}{\sqrt{n}} \)

 

The first step is to plug the table values into your calcuator. If you are not given the difference, enter each subject's scores into List 1 and List 2 (making sure to keep them matched!), and then plug L1 - L2 into List 3 in order to compute the difference. Then, take the list with the difference values and run them through 1-Var Stats. The mean your calculator computes from the difference values will be your point estimate, \(\bar{x}_d\), and the standard deviation it computes will be your \(s_d\) value.

For our specific problem, let's say we ran all the values through our calculator and found \(\bar{x}_d = 0.682\) and \(s_d = 1.84\).

→  \(\bar{x}_d \pm t^{*}\frac{s_d}{\sqrt{n}} \)  

→  \(0.682 \pm 1.306\frac{1.84}{\sqrt{36}} \) 

→  \(0.682 \pm 0.401\)

0.682 - 0.401 = 0.281

0.682 + 0.401 = 1.083

Interval: (0.281, 1.083)

 

Note: The critical value was found using a t-table. For t*, the critical value changes depending on the degrees of freedom. Remember:

For the t-distribution, degrees of freedom = \(n - 1\)

A portion of the t-table is listed below with the part needed for our problem highlighted:

 
df = 30 1.310 1.697 2.042
df = 35 1.306 1.690 2.030
df = 40 1.303 1.684 2.021
  80% 90% 95%

 

Step 4: State the Conclusion

Based on the data, I am 80% confident that the mean difference (coke - pepsi) in taste is between 0.281 and 1.083.