A one-sample t-test for the mean is a type of hypothesis test that is used with quantitative variables to determine whether a sample comes from a population with a specified mean.  Sometimes the population mean is not known, so it will instead be a hypothesized mean.  It compares the difference between the observed statistic and a hypothesized value to the standard error of the observed statistic.  This can be expressed with the equation: T=(Xbar-Mu) / (S/sqrt(N)).

The assumptions and conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval: random, 10%, and nearly normal.  If the conditions are met, we can proceed with the test.  

The t-value can be interpreted the same as a p-value.  If t< 0.05, we reject the null hypothesis, and accept the alternate hypothesis.  Therefore, if t > 0.05, we fail to reject the null hypothesis, and reject the alternate hypothesis because we have insufficient evidence to suggest otherwise.  

Example:

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

H0 :  Mu = 120      (This reflects our hypothesis that the mean of the yogurts calories will be 120)

HA:  Mu < > 120      (This reflects our alternative hypothesis that the mean will not fall at 120 calories)

Check conditions:  Random, 10%, and nearly normal (if sample size is less than 40, check QQ-plot/Histogram)

Mechanics:

We find and interpret the t-score

T=(Xbar-Mu) / (S/sqrt(N))

T=(131.82-120) / (25.226/sqrt(11))

T=1.554

Because the t-score is greater than 0.05 we fail to reject the null hypothesis that the mean vanilla yogurt will have a calorie count of 120.  We also do not have sufficient evidence to prove our alternative hypothesis that it does not equal 120, so we reject the alternative hypothesis as well.

SPSS can also perform these same mechanics, with less room for error.  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, hover over "Compare Means" to see a drop down menu, 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 hypothesized value for your data, because this is the value that is being testing.
• Do not change any other options.
• Click "OK".
• The "One-Sample Statistics" chart and "One-Sample Test" chart will appear in the output window.
• In the "One-Sample Test" chart we can see our t value.  For the above example, t = 1.554. 

The following video illustrates these steps: