A one-sample t-test for the mean is a type of hypothesis test that is used with one quantitative variable to ascertain whether the population is likely to have a specified mean. The population mean is unknown, so it will instead be a hypothesized mean.  

The equation above expresses that the test compares the difference between the observed statistic and a hypothesized value to the standard error of the observed statistic.  The assumptions and conditions that must be checked before the test can be run are:  random, 10%, and nearly normal.   You need to make sure that the sample is random and if you don't know if it is random, check if it could be representative.  Then check that the sample size is less than 10% of the population, and that the distribution is nearly normal.  The nearly normal condition can be checked by looking at the QQ-plot/histogram, or if the sample size is greater than 40.  If the conditions are met, we can proceed with the test.  

The t-value can be interpreted the same as a z-score to help create the model.  The t-value will tell you how many standard deviations out to go, then look at the alternative hypothesis to find out where the model needs to be shaded.  If it is a two-sided test, shade both portions to the left and right of the t-value.  If it is a one-sided test, only shade according to the alternative hypothesis.

For example, a nutritional guide book has claimed that the mean calorie content of Plain Vanilla Yogurt available in stores is 120 calories.  Accordingly, we have collected some samples from the local grocery store to see if this is true. So we set up our hypotheses like so:

     (This reflects our hypothesis that the mean of the yogurts calories will be 120)

      (This reflects our alternative hypothesis that the mean will not fall at 120 calories)

We must first check that all conditions are met.  We are not directly told that the collected yogurt samples are random; however, we will trust that the testers did their job correctly.  It is acceptable to say that the sample would be representative of all Plain Vanilla Yogurts.  Next, we check the 10% condition. 11 Vanilla Yogurts are surely less than 10% of all Vanilla Yogurt brands available, so this condition is met.  Finally, to check the nearly normal condition we check the Histogram/QQ-plot in SPSS because our sample is not greater than forty, our rule of thumb.  

After we find that our conditions for the t-test are met, we can continue by finding and interpreting the t-score.  We use the formula below where x-bar is the observed mean from the sample.  When calculated this number = 131.81.  Mu is the null hypothesis/test value.  In our case it would be the 120 calorie mean as stated in the nutritional guide.  S is the standard deviation, this is found in our output in SPSS and N is the sample size.  We have 11 different Plain Vanilla Yogurts, so this is our sample size.

With a t-value of 1.554, we know to shade the model 1.554 standard deviations from the mean on both sides of the model because we are doing a two-tailed test.  The mean of the model is the test statistic (null hypothesis) for our example, this will be 120.  

SPSS can also perform these same mechanics.  The output below shows the same t-score of 1.554.

Generating a one-sample t-test for the mean in SPSS

• Go to the "Analyze" menu, "Compare Means" and select "One-Sample T Test". 
• Drag the quantitative variable that you are testing to the "Test Variable(s)" box.
• Change the "Test Value" to the null hypothesis value for your data.
• Click "OK".  The One-Sample Statistics chart and One-Sample Test chart will appear in the Output window. 

The following video illustrates these steps: