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# Paired t-interval

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on December 1, 2014 at 2:30:52 pm

When comparing two independent sets of data, you subtract one from the other to find the difference for every group in the set. This doesn't make sense in general. It only works for certain kinds of data. These differences are the means of paired data. Makes no sense. After finding the mean differences, we want to know how confident we are of capturing the true difference. A paired t-interval constructs a confidence interval to estimate the mean difference between matched pairs of data. This confidence interval tells us how far away we are from the true mean. Incorrect. 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 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, plan. State what we want to know. 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 average high temperatures in January and in July from 12 different countries (Vienna, Copenhagen, Paris, Berlin, Athens, Rome, Amsterdam, Madrid, London, Edinburgh, Moscow, and Belgrade, respectively), we notice that one is a winter season and the other is a summer season.

January: 34, 36, 42, 35, 54, 54, 40, 47, 44, 43, 21, 37

July: 75, 72, 76, 74, 90, 88, 69, 87, 73, 65, 76, 84

Our plan is that 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 unimodal and symmetric.

Vienna: 75-34= 41

Copenhagen: 72-36= 36

Paris: 76-42= 34

Berlin: 74-35= 39

Athens: 90-54= 36

Rome: 88-54= 34

Amsterdam: 69-40= 29

London: 73-44= 29

Edinburgh: 65-43= 22

Moscow: 76-21= 55

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 90% confidence level.

36.8333+1.796(8.66375/3.4641)= 36.8333+4.491796= (32.3,41.3)

For our conclusion, we are 90% 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

• First open the file that you will be working with on SPSS We're assuming this has already been done. (By the way, each step is a sentence and, therefore, requires a period.)
• Select the Analyze drop down menu and choose Compare Means
• Under the Compare Means menu, select Paired-Sample T Test
• A menu will pop up, place the two different variables into the variable 1 and variable 2 boxes. Run-on sentence.
• 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
• 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

Again, you can assume the data set is open already. As you work through the steps, explain a little about the example. So introduce the data and make sure the viewer understands what they're looking at.

You can check one of the conditions in SPSS by calculating the differences in a new column and looking at a histogram and QQ-plot. It would be wise considering that you only have 12 data points.

You need to explain the choice you made to put July first and then January. The viewer will want to know how to make that decision and why one choice might be better than the other.

You need to help the viewer interpret the interval they see. What do those numbers mean in context?