Statistical Analysis for Education and Psychology Researchers

the logic of hypothesis testing not too demanding. The thinking underpinning the logic of
hypothesis testing is best illustrated with a specific example.

Example 4.10: Hypothesis Test about Proportions

Teachers in a Local Education Authority claim that the Authority is saving money by
limiting the number of statemented children. They believe that the proportion of referrals
that result in statements (an obligation for the Authority to provide special provision for a
child) has reduced drastically and is now lower than the national average. As principal
psychologist, you are sceptical about the teachers’ claims and decide to investigate them.
You believe that the number of statements issued would have been much higher than
they were if it were not for the new policy of supporting ordinary schools and thereby
increasing their capability to manage pupils who have special needs.
Your working assumption is that the proportion of referrals that lead to statements will
be no different from the national average. According to the Audit Commission Report,
Getting in on the Act, in England and Wales about 13 per cent of referrals result in a
statement being issued (Audit Commission Report, 1992). You decide to take a simple
random sample of referrals over the last two years and identify the proportion that have
resulted in statements being issued. The figure you obtain is 10.92 per cent.
The research question that you want to answer is: ‘Is the proportion of referrals
leading to statements in the local education authority statistically less than the proportion
nationally (13 per cent)?’ You begin the statistical investigation with a proposition or
hypothesis; The proportion of referrals leading to statements in the LEA is equal to 13
per cent (the national figure).
This is a hypothesis of no difference, that is the proportion of referrals leading to
statements in the LEA is the same as the national proportion. The hypothesis of no
difference is also called a null hypothesis or a statistical hypothesis. The null
hypothesis is a statement about the value of the parameter, π for the LEA population. If
the null hypothesis is true, then the proportion of referrals leading to statements in the
LEA would be the same as that throughout England and Wales. The null hypothesis is not
usually explained in words in research journals, rather a special notation is used: H 0.
The statistical or null hypothesis is held to be tenable until such times as data
collected from a sample yields results which suggest that it is no longer reasonable to
believe the null hypothesis. Clearly there is a true value for π, the proportion of referrals
leading to statements in the LEA and if the null hypothesis is true, this true value would
equal 13 per cent. If the null hypothesis is not true, then the laws of logic suggest that π
for the LEA population proportion of statements must equal some other value. Put
another way if the null hypothesis is not true then some other alternative hypothesis
must be true, the problem is we cannot know precisely what this alternative is. All we can
say is that it is not 13 per cent.
Thus far three important ideas in hypothesis testing have been introduced. The
statistical or null hypothesis of no difference, the alternative hypothesis, and the logic

that connects them both. The law of logic says that if the null hypothesis is not true then
some alternative hypothesis must be true.

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