Paired T-Interval

When comparing two sets of data, you subtract one from the other to find the difference for every group in the set. These differences are the means of paired data. This is called the Paired T-Test. After finding the mean differences, we want to know how close to the true mean we are for the matched pairs. A Paired T-Interval constructs a confidence interval to estimate the mean difference between matched pairs of data. This confidence level tells us how far away we are from the true mean. Before we can create a confidence interval, we need to check our conditions.

Conditions

Paired Data Condition - the data must be paired.

Independence Assumption - if the data are paired, the groups are not independent. Rather, the differences must be independent of each other.

Randomization Condition - the groups must be random.

Nearly Normal Condition - assume the population of differences follows a Normal Model. We can check this condition with a histogram.

Confidence Equations

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

                                                                                          dbar + t*_n-1 X SE(dbar)   where dbar is the mean difference,

                                                                                                                              t* is the critical value,

                                                                                                                              n-1 finds the degree of freedom, and

                                                                                                                              SE is the standard error

The standard error of the mean difference is:                          SE(dbar) = s_d /sqrt(n)   where s_d is the standard deviation of the differences, and

                                                                                                                             n is the sample size

Steps for Finding the Confidence Interval

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

Plan - state what we want to know.

Model - check the conditions, draw a picture, state the sampling distribution model, and choose your method.

Mechanics - estimate the standard error of dbar, and calculate the margin of error.

Conclusion - interpret the confidence interval in context.

Example

Looking at married couples, husbands tend to be slightly older than wives. How must older, on average, are husbands? We have data from a random sample of 200 British couples. Only 170 couples provided ages for both husband and wife. Form a confidence interval for the mean difference of husband's and wife's ages for these 170 couples.

Plan - I want to estimate the mean difference in age between husbands and wives. I have a random sample of 200 British couples, 170 of whom provided both ages.

Model - Paired Data Condition: the data are paired because they are testing on members of married couples.

         - Independence Assumption: the data are from a randomized survey, so couples should be independent of each other.

         - Nearly Normal Condition: a histogram is best.

         The conditions are met, so we can use Student's t-model with (n-1)=169 degrees of freedom and find a paired t-interval.

Mechanics - n=170

               - dbar=2.2

               - s_d=4.1 years

               - standard error of dbar: SE(dbar) = s_d /sqrt(n)

                                                                = 4.1/sqrt(170) = 0.31 years

                                                                degree of freedom(df): n-1=169

                - the 95% critical value for t₁₆₉ is 1.97

                - the margin of error: ME = t*₁₆₉ X SE(dbar)

                                                     = 1.97(0.31) = 0.61

                So the 95% confidence interval is 2.2 + 0.6, or an interval of (1.6,2.8) years

Conclusion - I am 95% confident that British husbands are, on average, 1.6 to 2.8 years older than other wives.

Using SPSS to find Confidence Intervals for Matched Pairs

• First open the file that you will be working with on SPSS
• 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
• 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