Download MTH 202 : Probability and Statistics

MTH 202 : Probability and Statistics
Lecture S3 :
12. Testing Hypothesis
12.1 : Hypothesis and errors
In statistical problem, we dealt with the estimation of parameters in
the last section. A comparative study of several parameters is often
necessary since we would not know for sure which one would be more
appropriate for the particular population data. Also the meaning of
the phrase ”more appropriate” need to be well understood. We will
start with the following definition :
Definition 12.1.1 : A statistical hypothesis is an assertion or conjecture about the distribution of one or more random variables. If a
statistical hypothesis completely specifies the distribution, it is referred
to as a simple hypothesis; otherwise, it is referred as a composite
hypothesis.
For example say we are given two dies and say p1 and p2 represents the
probability of getting a head on one toss with these dies respectively.
Suppose we also know the information that p1 = 1/6 and p2 is much
larger than 1/6 (I presume some of you follow the ongoing series of
Mahabharata and remember the tricked die of Shakuni). Next a die is
given to you without telling the corresponding value of pi and you are
asked to find out the value by some series of experiments. Towards this
you would make a conjecture H that the probability p of your die is
1/6 (which also tells about the distribution since it is binomial). Then,
H is a simple hypothesis.
Thus a simple hypothesis specifies the functional form of the distribution as well as the parameters. Thus while building up a statistical
data model, often a simple hypothesis is put forward which need to be
further analyzed. This in standard terminology, referred as the null
hypothesis and denoted by H0 . To compare this with a different set
of parameters, an alternate hypothesis (denoted by H1 ) is put up
which can either be simple or composite.
1
2
While we develop a method to compare these, we may come across two
kinds of error.
Definition 12.1.2 :
Type-I error : we reject the null hypothesis H0 while it is true. The
probability of committing type-I error is denoted by α.
Type-II error : we accept the null hypothesis H0 while it is false.The
probability of committing type-II error is denoted by β.
The rejection region for H0 is called the critical region.
TO BE UPDATED
References :
[RS] An Introduction to Probability and Statistics, V.K. Rohatgi and
A.K. Saleh, Second Edition, Wiley Students Edition.
[RI] Mathematical Statistics and Data Analysis, John A. Rice, Cengage
Learning, 2013