This article is the second in a four part series on essential statistical techniques for any scientist or engineer working
in the biotechnology field. This installment presents statistical methods for comparing sample means, including how to establish
the correct sample size for testing these differences. The difference between one-sample, two-sample, and z-test also are
In hypothesis testing, we must state the assumed value of the population parameter called the null hypothesis. The goal of
hypothesis testing is to verify if the sample data is part of the population of interest. You either have sufficient evidence
to accept the null hypothesis or reject it—you do not prove it. The significance level or p-value indicates the likelihood
that the sample comes from the population of interest. Statisticians usually use a p-value of 0.05 as the cutoff for statistical
significance. In other words, a p-value less than 0.05 is sufficient evidence to reject the null hypothesis. Typically, the
null hypothesis is a statement about the value of the population parameter. For example, μ = 100 versus μ ≠ 100. A one-sided
test means we are testing the null hypothesis of either less than or greater than. A two-sided test means we are testing the
null hypothesis of less than and greater than.
The one-sample t-test is used to compare a sample mean to a hypothesized population mean. The hypothesis can be either a one-sided
or two-sided test. Usually, the population variance is unknown requiring use of the t-distribution, which takes into account
the uncertainty in estimating the sample variance. The t-distribution is tabled by confidence level and degrees of freedom.
For the one-sample t-test, the degrees of freedom are the number of observations used to estimate the sample standard deviation
minus one. The formula for the one-sample t-test is as follows:
in which X-mean is the sample mean, μ is the theoretical population mean, s is the sample standard deviation, and n is the
sample size used to estimate the mean and standard deviation.
If the value of t* is greater than the tabled value from the t-distribution, the sample mean is statistically different than the population
mean (μ). An example of a one-sample t-test would be comparing protein concentration for a particular batch to a theoretical
protein concentration. Table 1 shows an example of a two-sided one-sample t-test for protein concentration. The hypothesis
is that the lot is not statistically different than 30 (μ = 30). The mean of the six vials was not statistically different
than the theoretical value of 30 (p = 0.223). The t* of 1.355 did not exceed the tabled value for a 95% confidence level with five degrees of freedom of 2.571.
Table 1. An example of a two-sided one sample t-test for protein concentration. The hypothesis is that the lot is not statistically
different than 30 (μ = 30).