Download Hypothesis Introduction

Hypothesis Testing
In Statistics 5 we considered methods of estimating an unknown parameter .
For example if we take a random sample X of size n from a population N(, 2), we
(n  1) 2
S respectively.
estimate the two unknown parameters  and 2 by X and
n
This is called point estimation for the parameters.
In the first chapter of Statistics 6 we developed methods for specifying an interval which we
are confident contains . This is interval estimation. Now we consider a different question
and start with a hypothesis.
Before examining the theory of hypothesis testing in detail, we will discuss a practical
application of the method.
Practical Example
Imagine that a company states in its sales pitch that a particular model of its mobile phones
lasts for 150 hours before it is required to be next charged up. If you were thinking of
buying one, you may like to obtain some proof that this assertion is true. One way of doing
this is to take a sample of phones, make a number of measurements and then calculate
the mean number of hours between charging. It would be impossible to do this for every
phone produced since the population is so large, so the best that can be done is to
calculate a sample mean.
Suppose that a sample of 40 was taken and this produced a mean value of 147.4 hours
(with a standard deviation of 12 hours). Does this mean that the manufacturer's claim has
been disproved? Clearly 147.4 is less than 150 so it looks as if the manufacturer could be
over-estimating the time between charging. However, it must be appreciated that this was
just one sample; if another sample was taken it might give a very different result (for
example, it could give a value of 152.3 hours, in which case the phones are doing better
than the manufacturers claim!).
The method of hypothesis testing starts by making an assertion about the population, often
an assumption that the mean is equal to a stated result. In this case it is hypothesised that
the population mean, μ, for the mobile phones is 150 hours.
Method of Solution
Assert that the population mean is what the company claims it to be. Call this assertion H 0
– the Null Hypothesis.
We carry out an analysis which will allow us to either accept this assertion as being true, or
to accept the decision that the population mean is not what the company claims it to be.
This can be written as:
H0:  = 150
H1:  ≠ 150
Using the theory of Chapter 1, assuming H0 to be true leads us to the conclusion that 95%
of samples of size 40 taken from the population must be approximately between
12
, i.e. from 146.28 to 153.72.
150  1.96
40
Note that since our sample mean, 147.4, lies within these limits, we accept the null
hypothesis. We would not be inclined to argue with the manufacturer’s claim.