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Comparing a Mean with a Standard

KEY WORDS t-test, hypothesis test, confidence interval, dissolved oxygen, standard.

A common and fundamental problem is making inferences about mean values. This chapter is about
problems where there is only one mean and it is to be compared with a known value. The following
chapters are about comparing two or more means.
Often we want to compare the mean of experimental data with a known value. There are four such situations:

1. In laboratory quality control checks, the analyst measures the concentration of test specimens
that have been prepared or calibrated so precisely that any error in the quantity is negligible.
The specimens are tested according to a prescribed analytical method and a comparison is made
to determine whether the measured values and the known concentration of the standard speci-
mens are in agreement.
2. The desired quality of a product is known, by specification or requirement, and measurements
on the process are made at intervals to see if the specification is accomplished.
3. A vendor claims to provide material of a certain quality and the buyer makes measurements
to see whether the claim is met.
4. A decision must be made regarding compliance or noncompliance with a regulatory standard
at a hazardous waste site (ASTM, 1998).

In these situations there is a single known or specified numerical value that we set as a standard against
which to judge the average of the measured values. Testing the magnitude of the difference between the
measured value and the standard must make allowance for random measurement error. The statistical
method can be to (1) calculate a confidence interval and see whether the known (standard) value falls
within the interval, or (2) formulate and test a hypothesis. The objective is to decide whether we can
confidently declare the difference to be positive or negative, or whether the difference is so small that
we are uncertain about the direction of the difference.

Case Study: Interlaboratory Study of DO Measurements
This example is loosely based on a study by Wilcock et al. (1981). Fourteen laboratories were sent
standardized solutions that were prepared to contain 1.2 mg/L dissolved oxygen (DO). They were asked
to measure the DO concentration using the Winkler titration method. The concentrations, as mg/L DO,
reported by the participating laboratories were:
1.2 1.4 1.4 1.3 1.2 1.35 1.4 2.0 1.95 1.1 1.75 1.05 1.05 1.4
Do the laboratories, on average, measure 1.2 mg/L, or is there some bias?

Theory: t-Test to Assess Agreement with a Standard

The known or specified value is defined as η 0 . The true, but unknown, mean value of the tested specimens
is η, which is estimated from the available data by calculating the average .y

© 2002 By CRC Press LLC