Small Sample t Test

For a small number of results, we use a similar procedure to the z test except that the test statistic is the t statistic. Again we test the null hypothesis H0: m = m0, where m0 is a specific value of m such as an accepted value, a theoretical value, or a threshold value. The procedure is:

As an illustration, consider the testing for systematic error in an analytical method. In this case, a sample of accurately known composition, such as a standard reference material, is analyzed. Determination of the analyte in the material gives an experimental mean that is an estimate of the population mean. If the analytical method had no systematic error, or bias, random errors would give the frequency distribution shown by curve A in Figure 1.1. Method B has some systematic error so that x̅B, which estimates mB, differs from the accepted value m0. The bias is given by

Bias = μ_B - μ₀                                         (1.1)

Figure 1.1 Illustration of systematic error in an analytical method. Curve A is the frequency distribution for the accepted value by a method without bias. Curve B illustrates the frequency distribution of results by a method that could have a significant bias due to a systematic error.

In testing for bias, we do not know initially whether the difference between the experimental mean and the accepted value is due to random error or to an actual systematic error. The t test is used to determine the significance of the difference. Example 7-5 illustrates the use of the t test to determine whether there is bias in a method.

Example 7-5

A new procedure for the rapid determination of sulfur in kerosenes was tested on a sample known from its method of preparation to contain 0.123% S (μ₀ = 0.123% S). The results for % S were 0.112, 0.118, 0.115, and 0.119. Do the data indicate that there is a bias in the method at the 95% confidence level?

Solution