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.

- Before calculating the confidence interval, you must use SPSS to check the nearly normal condition.
- Select Transform, then Compute Variable. Name the Target Variable "D" (for the difference between the pairs) and calculate July - Jan to find that difference.
- Select Analyze, Descriptive Statistics, Explore then place the new D variable in the dependent list. Change the plots to include histogram and normality plots.
- This SPSS output includes the Q-Q Plot to show that it meets the nearly normal condition.
- To find the confidence interval, select the Analyze drop down menu and choose Compare Means.
- Under the Compare Means menu, select Paired-Sample T Test.
- The Paired-Sample T Test window will pop up. Place the two different variables into the variable 1 and variable 2 boxes.
- SPSS has the confidence interval set to 95%, but if you would like to change the interval select the options button and replace the 95 with the percentage that you want.
- After you have replaced the 95% with your intended interval click continue to return to the Paired-Sample T Test menu.
- Select the OK button and SPSS will produce the output for the Paired-Sample T Test.
- Scrolling through the output, in the Paired Samples Test box there is a column for the interval that you specified that breaks it into the upper and lower ends of the interval.