From Paired Differences to a Confidence Interval
In “Checking Normality of the Differences,” you learned to graph the paired differences when assessing the shape condition. Here, we use those differences to estimate a population mean difference. The paired t confidence interval is built from the sample mean difference, \(\bar d\), and the sample standard deviation of the differences, \(s_d\).
The key is to analyze one difference per pair, using the subtraction order established for the study. For example, if \(d_i=\text{after}_i-\text{before}_i\), then a positive difference means the after measurement was higher than the before measurement. The parameter \(\mu_d\) is the true mean of these differences for the population of pairs named in the study.
This is a one-sample t interval applied to the list of differences. The sample size \(n\) is the number of pairs, not the number of individual measurements in the two original columns. The interval is centered at \(\bar d\), and its endpoints are \(\bar d\) minus and plus the margin of error.
To find \(t^*\), let \(C\) be the confidence level written as a decimal and let \(\alpha=1-C\). The interval leaves an area of \(\alpha/2\) in each tail of the t distribution. Thus, the area to the left of the positive critical value is \(1-\alpha/2\). On a calculator, use \(\text{invT}(1-\alpha/2,n-1)\). For example, a 95% interval has \(\alpha=0.05\), so the left-tail area for \(t^*\) is \(0.975\).
Use the stated subtraction order, and identify \(\mu_d\) in the context of the target population.
Consider the paired design, random selection or an appropriate randomized process, independence between pairs, the 10% condition when applicable, and the shape of the differences.
Use \(df=n-1\), the chosen confidence level, and \(SE=s_d/\sqrt{n}\).
Subtract the margin of error from \(\bar d\) for the lower endpoint and add it for the upper endpoint. Keep the units of the differences.
As in “Conditions for a Paired t Procedure,” the design and independence checks matter as well as the graph of the differences. A calculation by itself does not establish that a t interval is appropriate. In the examples, each condition check is stated before the interval is constructed.
Worked Examples
Worked Example: Flexibility Changes After a Stretching Routine
A fictional recreation center takes a random sample of 8 participants from 600 enrolled adults and records each participant’s reach distance before and after a stretching routine. Define \(d_i=\text{after}-\text{before}\), in centimeters. The differences are \(2,3,4,5,5,6,7,8\). A graph of the differences is roughly symmetric, with no pronounced outlier. Construct a 95% confidence interval for the population mean difference for enrolled adults.
State. Let \(\mu_d\) be the true mean after-minus-before change in reach distance, in centimeters, for all enrolled adults. We want a 95% confidence interval for \(\mu_d\).
Plan. The same participants are measured twice, so these are paired data and we use the 8 within-participant differences. The participants were randomly sampled. The 8 participants can reasonably be treated as independent of one another, and \(8<0.10(600)=60\), so the 10% condition is met. The sample is small, but the graph is described as roughly symmetric with no pronounced outlier, which supports the shape condition. These checks support using a paired t interval.
Do. The differences have sum \(40\), so \(\bar d=40/8=5\) cm. Their deviations from 5 are \(-3,-2,-1,0,0,1,2,3\), and the sum of squared deviations is \(9+4+1+0+0+1+4+9=28\). Therefore,
For a 95% interval, \(\alpha=0.05\), so \(t^*=\text{invT}(0.975,7)\approx2.365\). The standard error and margin of error are
Thus, the interval is
or approximately \((3.33,\ 6.67)\) cm.
Conclude. The constructed 95% paired t interval for the mean after-minus-before change in reach distance among enrolled adults is approximately 3.33 to 6.67 centimeters. It is centered at the sample mean difference of 5 centimeters.
Worked Example: Room Temperatures After Insulation
A fictional building study takes a random sample of 16 rooms from 400 rooms in similar buildings. Each room’s temperature is measured before and after insulation under comparable conditions. Define \(d_i=\text{after}-\text{before}\), in degrees Celsius. The study reports \(\bar d=0.8\) degrees, \(s_d=1.6\) degrees, and a roughly symmetric distribution of differences with no pronounced outliers. Construct a 90% confidence interval for the population mean difference.
State. Let \(\mu_d\) be the true mean after-minus-before temperature difference, in degrees Celsius, for rooms in the target population. We seek a 90% confidence interval for \(\mu_d\).
Plan. The same room is measured twice, making this a paired design. The rooms were randomly sampled, and the 16 sampled rooms can reasonably be treated as independent. The 10% condition holds because \(16<0.10(400)=40\). The reported shape of the differences supports the shape condition. We can use a paired t interval.
Do. Here \(n=16\), so \(df=16-1=15\). For a 90% interval, \(\alpha=0.10\), giving a left-tail area of \(1-\alpha/2=0.95\). Thus, \(t^*=\text{invT}(0.95,15)\approx1.753\). The standard error and margin of error are
The interval is
which rounds to approximately \((0.10,\ 1.50)\) degrees Celsius.
Conclude. The 90% paired t interval for the population mean after-minus-before room temperature difference is approximately 0.10 to 1.50 degrees Celsius. The calculation uses the room-level differences, not two separate samples of room temperatures.
Worked Example: Soil pH After an Amendment
A fictional community garden project randomly selects 25 paired plots from 500 plots. Each plot’s soil pH is measured before and after an amendment. Define \(d_i=\text{after}-\text{before}\). The differences have sample mean \(\bar d=0.6\) pH units and sample standard deviation \(s_d=1.25\) pH units. A graph shows no strong skewness or pronounced outliers. Construct a 99% confidence interval for the population mean difference.
State. Let \(\mu_d\) be the true mean after-minus-before change in soil pH, in pH units, for plots in the target population. We seek a 99% confidence interval for \(\mu_d\).
Plan. Each plot has both measurements, so the data are paired. The plots were randomly selected, and distinct plots can reasonably be treated as independent. The 10% condition is satisfied because \(25<0.10(500)=50\). The graph of the differences provides reasonable shape support. These conditions support a paired t interval.
Do. The degrees of freedom are \(25-1=24\). A 99% interval has \(\alpha=0.01\), so the area to the left of the positive critical value is \(0.995\). The calculator gives \(t^*=\text{invT}(0.995,24)\approx2.797\). The standard error and margin of error are
The interval is
or approximately \((-0.10,\ 1.30)\) pH units.
Conclude. The constructed 99% paired t interval for the population mean after-minus-before soil pH change is approximately \(-0.10\) to \(1.30\) pH units. Its endpoints are calculated from the observed mean difference and the margin of error for the requested confidence level.
Common Mistakes and AP Exam Tips
A correct paired t interval depends on using the right data, critical value, and degrees of freedom. Keep full precision through the calculation and round the final endpoints at the end. If you round the standard error early, the resulting margin of error and endpoints may differ slightly from the calculator result.
- Using two-sample methods on paired data. Measurements from the same person, object, or matched pair are linked. First compute one difference per pair; then use \(\bar d\) and \(s_d\).
- Forgetting the subtraction order. If differences are defined as after minus before, the interval estimates the mean after-minus-before difference. Reversing the order changes the signs of the mean and both endpoints.
- Using the number of measurements instead of the number of pairs. With 16 pairs, \(n=16\), not 32. The degrees of freedom are \(16-1=15\).
- Choosing the wrong t critical value. Use the confidence level to split the remaining area equally between the two tails. For a 90% interval, use the 0.95 left-tail area; for a 99% interval, use the 0.995 left-tail area.
- Using \(s_d\) as the standard error. The standard error is \(s_d/\sqrt{n}\). The margin of error is \(t^*\) times that standard error.
- Skipping the conditions. A complete response connects the paired design, random selection or randomization, independence, the 10% condition when relevant, and the shape of the differences to the study facts.
- Reporting only a formula. Substitute the actual values, show the degrees of freedom and critical value, calculate both endpoints, and retain the units of the differences.
For full-credit communication, identify \(\mu_d\) in context, state the confidence level, justify the paired t procedure with specific condition evidence, and show the interval calculation. In the next tutorial, the focus will be how to interpret the completed interval carefully in context.
Check Your Understanding
Use the paired differences and confidence level in each question to plan or calculate an interval.
- A study has 12 pairs. What degrees of freedom should be used for its paired t interval?
- For an 80% confidence interval, what area to the left of the positive critical value is needed?
- A sample has \(\bar d=-2.4\), \(s_d=3\), and \(n=9\). Write the expression for the standard error.
- Why should a paired t interval use one difference per pair rather than treating the two measurement columns as independent samples?
- If the differences are defined as after minus before, what does a negative interval endpoint indicate about the direction of that endpoint relative to zero?