Paired t-test

A paired t-test compares means from within a group to help determine weather we can accept or reject the null. It is often used when comparing a sample groups scores before and after. 


A t-test a type of hypothesis test. First you would decide what the population of the test is. The next step in this test would be to state the null and alternate hypothesis. The equations listed below will be your equations to determine a your null and alternate hypothesis. However you need to be sure to write out the null and alternate hypothesis in full sentences as well.  

null hypothesis:  .  The null is stated as the average of the means or μ. 

alternate hypothesis     The alternate hypothesis is the average of the means of what we are trying to prove. 

The next step would be to check the conditions the conditions to be checked are as follows. As before make sure to write the conditions in full sentences. 

1. Is the sample random, if you cant determine this then ask yourself if its representative of the population.

2. Independence condition

3. Nearly normal condition

The next step you would take would be to calculate your test statistic, if by hand the equation is listed below. 


After this you would draw out your t-model based off of your information. 

Depending on results from your test you would either choose to reject the null value if there is sufficient evidence that your null value is untrue. If you are not able to reject the null value you would say you have insufficient evidence to reject the null. After this you would state your conclusion in a contextually meaningful way.  


Now let us do an example, the example we will be using is the one shown in the video below where we have some cities from Europe and we want to see if the temperatures in January are different than the ones in July. Our null hypothesis would be that the temperatures are the same, and our alternate hypothesis is that they are different. 

The next step we would take would be to check our conditions. 

Since we have no information on how the cities were selected we can not say that it is random but we hope that the cites are representative. 

We know these cities are independent because the temperature of one does not affect the temperature of another. 

Since this is a small data set we must check the nearly normal condition to check this we must look at the histogram and qq plot in SPSS.



We can see that the Histogram and QQ Plot meet our conditions for normality (Linear/Unimodal) and we can proceed with our test. 


We would next compute our test statistic or t-value using either the formula or SPSS, in SPSS it can be found here and we can see that it is -14.727, we can also find our p-value here as well and it is listed in the sig. column, we can see that it is listed as .000. While it says it is 0 that is not true it is just to small to be displayed. 

From this we can reject the null since there is sufficient evidence that the temperature in July is higher than the temperature in January. 



 To conduct a Paired Sample T-Test in SPSS follow these simple steps:



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