Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

xiii

Introduction to Hypothesis Testing

Hypothesis testing is the fundamental process of statistical inference. It is in

fact the backbone of the scientic method. Unfortunately, it is also the most

misunderstood, misapplied, and misinterpreted of all statistical concepts.

Aside from the mathematical formalism of hypothesis tests, there are sev-

eral ideas that make hypothesis testing the central component of empirical

investigation.

1. Data are based on a subset (sample) of individuals from a larger,

often innite population.

2. The measurements or quantied observations made on the individ-

uals are associated with a probability distribution, which has some

parameters such as the mean, standard deviation, and percentiles.

3. The only information we have (presuming no a priori knowledge

of the parameter) about the values of these parameters comes from

the sample statistics calculated using the measurements or observa-

tions made on individual members of the sample, such as the sample

mean, the sample standard deviation, and the sample percentiles.

4. If we obtained two samples from the same population, each sample

having the same number of individuals, and computed the sample

statistics for some measurement, the two samples would very likely

yield different values for the same statistic. Neither sample is likely

to yield the actual value of the corresponding population parameter.

5. The number of individuals in the population is too large to feasibly

observe every single individual; in other words, we MUST rely on

sample information to determine anything about the population

parameters.

The most important idea that makes hypothesis testing so indispensable is

6. It is desirable to demonstrate, using empirical observations, that

some parameter or set of parameters falls into some particular range

of values.

Because we understand from item number 4 that any sample statistic is

subject to sampling variation, we must couch our conclusions about popula-

tion parameters in the language of probability. Furthermore, we must be able

to state a priori into what range we are hoping to demonstrate the param-

eters fall.

xiv Introduction to Hypothesis Testing

Formal Hypothesis Testing

Every hypothesis test begins with a statement of two hypotheses, or asser-

tions, about some parameter or set of parameters that are the logical negations

of each other. The rst, by convention, is often called the “null” hypothesis

and is labeled H

0

. The alternative, which is the logical negation of the null

hypothesis, is symbolized in this work by H

1

. The null hypothesis is gener-

ally a statement of something we hope is not true. The idea is that, using

data, we hope to reject the null hypothesis. We reject the null hypothesis if

it is not likely enough to have obtained the observations we made given that

the null hypothesis is true. “Likely enough” is customarily dened as a prob-

ability greater than 5percent. Typically, the null hypothesis is stated in such

a way as to make it possible to assess the limits to a feasible range for sample

statistics given that the null hypothesis is actually true. Note that in elemen-

tary statistics texts the null hypothesis is almost always stated as an equality,

either the value of a parameter between two or more groups, or the equality

of a parameter for a single population to a particular value. In the case of

equivalence and noninferiority, the null is usually stated as an inequality,

the nature of which is always adverse to the experimenter’s general desire.

As an example, consider the hypotheses

H

0

: μ = 10

and its logical negation:

H

1

: μ ≠ 10

Here, the Greek letter μ symbolizes the mean value of some measurement

made on individuals, averaged over all individuals in the population. Note

that it is not possible to calculate the arithmetic average of any measure-

ment for a population with an innite number of members, and it may not

be economically feasible for a population with a nite number of members.

Without elaborating, let us just assume that some population mean exists,

but its value is unknown. Suppose further that we sample n = 15individuals,

and compute the sample mean and the sample standard deviation:

X 9.3=

S = 1.7

So, do we believe that the population mean is in fact 10 or not? After all, 9.3

doesn’t seem too far off from 10. Perhaps if we sampled again, we might get

a different sample mean that was greater than 10. Perhaps we cannot afford

to sample more than these 15individuals. Statistical theory tells us how to