In Chapter 7, you learned how to use the normal (or Gaussian) distribution, which is a continuous probability distribution, to assist in making inferences about a population.Recall that the known mathematical properties of the distri- bution can be used to determine probabilities of characteristics occurring within the population, even when the population mean is unknown.Thus, hypothesis testing can be carried out using limited sampling, and correct inferences drawn, if the population is normally distributed.In many natural systems, populations are normally distributed, but sometimes they are not, and thus, the normal distribu- tion cannot be used as a model.
However, if you have gathered enough samples, it may still be possible to use the properties of the normal distribution, since the sampling distribution of averages is likely to be normal, according to the Central Limit Theorem (or at least have some of the key characteristics of a normal distribution, such as being unimodal and symmetrical).Thus, irrespective of the underlying population distribution, the normal distribution can be used to estimate probabilities when samples are sufficiently large: the sample variance can be used to estimate the population vari- ance, and inferences drawn with the assistance of the normal distribution.
This strategy may not always be appropriate for answering your specific research question, especially if you can only obtain limited samples because of financial, physical, or time constraints.Indeed, such a situation faced industrial statistician William Gosset in the early twentieth century, when he worked at the Guinness Brewery in Dublin as an industrial researcher with an enviable role—quality assurance for beer brewing.After studying statistics with Karl Pearson at Univer- sity College, London, Gosset published a paper under the pseudonym “Student,” since Guinness did not want their competitors to know that they were employing statistics to improve quality control.
Gosset’s key observation was the dependence on sample size for determining the probability that the mean of the population lies within a given distance of the mean of the sample, if a normal distribution is assumed.Through a combination of mathematical argument and numerical simulation, Gosset noted that when samples are collected from a normal distribution, and if the number of samples is small, and these are used to estimate the variance, then the distribution (for the variablex):
is both flatter, and has more observations appearing in the tails, than a normal distribution, when sample sizes are less than 30, and where refers to the standard error.Since bothsandxare random variables, this may not be such a surprise.However, as the number of samples increases, the distribution becomes normal, given the dependence on n, and the corresponding effect on degrees of freedom, since df =n – 1.This distribution is known as the t distribution, and approximates a normal distribution ifn(and by implicationdf) are large (>30 in practical terms).
Books of statistical tables normally provide critical values oftthat can be used at different degrees of freedom to make inferences about the population, with an asso- ciated probability of committing a Type I error (α).For example, wheren= 21 and df= 20, thent= 1.725 at thep= 0.05 significance level, andt= 2.528 at thep= 0.01 significance level.These relations would usually be expressed ast0.05,20= 1.725 and t0.05,20= 1.725, respectively. Figure 8-1 shows an exampletdistribution fordf= 5, 15, and 25, compared to a normal distribution.
t-Tests
Now that you have seen what thetdistribution is, you may be wondering about its purpose.In simple terms,t-tests are the simplest form ofparametric hypothesis
testingfor real-valued (rather than categorical) data.Using a t-test allows you to
test whether the mean of a sample differs significantly from an expected value, or whether the means of two groups different significantly from each other.Signifi- cance here means statistical significance, and is related to the probability (p) of committing a Type I error.Typically acceptable probability values arep< 0.05, representing a 1 in 20 chance of committing a Type I error, orp < 0.01, repre- senting a 1 in 100 chance of committing a Type I error.
t x–µ s n --- --- = s⁄ n
t-Tests | 153
The t-Test
The probability of committing a Type I error relates to two different ways that hypothesis testing is used: in science and in technology.Scientists typically frame their experiments so that they do not directly test the hypothesis, but evaluate a
null hypothesis.Thus, Type I error here applies to the probability of rejecting the
null hypothesis, when in fact it should have been accepted.For example, a scien- tist has formed two groups, treatment and control, testing the effects of a new weight-loss drug, and her hypothesis is that the weight-loss drug will significantly reduce weight in her treatment group.Weight is measured pre-test and post-test (after six weeks of taking the drug).Age, sex, height, and weight-matched partici- pants are randomly allocated to the treatment and control groups.The null Figure 8-1. Comparison of the normal and t distribution for v = 5, 15, and 25