Page 47 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing

5.1 Hypothesis Tests

In the last chapter we have discussed an important statistical inference: estimation of parameters, where
our objective is to estimate the unknown true value of a parameter. In this chapter we introduce another
important type of statistical inference: testing of statistical hypothesis, where we shall be interested to examine
whether the data from a random sample support or refute a conjecture about the true value of a parameter.
As an example, the manufacturer of a certain water filter may claim that the filtered water, has an average,
a pH value of 6.4. As another example, an insurance company may claim that the percentage of population
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who buy health insurance this year has increased compared to last year’s 12%. In the first example, it is
to test a hypothesis about the population mean. In the second example, it is to test a hypothesis about the
population proportion.
The truth or falsity of a statistical hypothesis is never known to us with absolute certainly unless we examine
the entire population. This, of course, would be impractical in most situations. Instead, we examine a random
sample from the population to produce evidence that either supports or refutes the hypothesis. The evidence
from the sample that is inconsistent with the stated hypothesis leads to the rejection of the hypothesis. A
hypothesis test or significance test is a method of using sample data as evidence to test a statistical hypothesis
about a population parameter.

The null and alternative hypothesis

The structure of hypothesis testing will be formulated with a null hypothesis denoted by H and an alternative
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hypothesis denoted by H . Usually H specifies a particular value for a population parameter, and H specifies
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a range of values. In general, the null hypothesis H represents there is no difference between the claim and
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reality whereas the alternative hypothesis H represents there is statistically significant difference between
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the claim and reality. In a hypothesis test, H is assumed to be true and information obtained for a sample
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statistic is used to determine whether there is strong evidence to reject H .
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Example 1
A hypothesis test is performed to determine whether the mean value of filtered water from a certain
type of water filter is 6.4.
State the null hypothesis and alternative hypothesis for the hypothesis test.
Solution: Let µ denote the mean pH value of filtered water.
The null hypothesis is that the mean pH value is 6.4, i.e. H : µ = 6.4
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5 The alternative hypothesis is H : µ ≠ 6.4
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Example 2

A hypothesis test is performed to determine whether the percentage of population who buy health
insurance this year has increased compared to last year’s 12%.
State the null hypothesis and alternative hypothesis for the hypothesis test.
Solution: Let p denote the proportion of population who buy health insurance this year.
The null hypothesis is that this year’s percentage remains the same as last year.
H : p = 0.12
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The alternative hypothesis is that this year’s percentage has increased.
H : p . 0.12
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Note: The null hypothesis H is usually stated using the equality sign.
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05 STPM Math(T) T3.indd 238 28/10/2021 10:24 AM