Paired t-interval

When given two sets of quantitative variables, we use a paired t-interval to construct a confidence interval to estimate the mean difference between a pair of data.  With these average values, we can find the mean difference of the quantitative data. After finding the mean differences, we want to know how confident we are of capturing the true difference. This confidence interval estimates at a certain confidence level where the true mean is.


 Before we can create a confidence interval, we need to check our conditions. The paired data condition states that the data set must be paired. Along with this condition, if the data are paired, the groups themselves are not independent, rather the differences must be independent of each other. This is the independence assumption. The randomization condition states that all groups must be random in the sample in order to run the test. The nearly normal condition assumes that the population of differences follows a normal model. This condition is met if the sample size is greater than 40 or it can be checked with a histogram or QQ-plot.


In order to conduct the test, we need to follow the appropriate steps to find a solution and come to a conclusion.

First, figure out what we want to know. Identify the population and sample for better understanding. Then, use model steps to check conditions, draw a picture, state which distribution model you will be using, and choose your method. Next, use mechanics to test the paired data. Estimate the standard error and calculate the margin of error. For your conclusion, interpret the confidence interval contextually.


If all of our conditions are met, we can calculate a confidence interval:

 where  is the mean difference and  is the critical value with  degrees of freedom.


For example, if we are given monthly average high temperatures in January and in July from each of 12 European cities (Vienna, Copenhagen, Paris, Berlin, Athens, Rome, Amsterdam, Madrid, London, Edinburgh, Moscow, and Belgrade), we notice that one is a winter season and the other is a summer season. Each city gets two variables: the average high in January and the average high in July.

We want to know the confidence interval for the mean temperature difference between the two seasons, since both are different extremes of each other. Next, we check our conditions. We see that these data are paired, average temperatures in January and average temperatures in July. The data from this random survey collected random temperatures throughout the month from twelve countries, so we know the temperatures should be independent of each other, meeting the independence assumption. We check this data set with a histogram to find it uni-modal and symmetric.


For this data set, after subtracting January from July, our mean difference (dbar) is 36.8333.

Since our conditions are met, we can model using a t-model with 11 (12-1) degrees of freedom.

For mechanics, n=12, and our average mean of 36.8333, we can calculate our confidence interval with a 95% confidence level.

36.8333+2.201(8.66375/3.4641)= 36.8333+4.491796= (32.3,41.3)

For our conclusion, we are 95% confident that the mean temperature difference between summer and winter is between 32.3 and 41.3 degrees.



Using SPSS to find Confidence Intervals for Matched Pairs




Unable to display content. Adobe Flash is required.