1305AFE Lecture 9: Week 9 Business Data Analysis Lecture Notes
Week 9 Business Data Analysis Lecture Notes
Statistical Inference: Hypothesis testing
Hypothesis testing: Describing a single population
Introduction
• The purpose of hypothesis testing is to determine whether there is enough statistical
evidence in favour of a certain belief about a population parameter.
• Example
o In a criminal trial, a jury must decide whether the defendant is innocent or
guilty based on the evidence presented at the court.
Concepts of hypothesis testing:
1. There are two hypotheses, the null and the alternative hypotheses.
2. The procedure begins with the assumption that the null hypothesis is true.
3. The goal is to determine whether there is enough evidence to infer that the
alternative hypothesis is true.
In a criminal trial
• A criminal trial is an example of hypothesis testing without the statistics.
• In a criminal trial, a jury must decide whether the defendant is innocent or guilty
based on the evidence presented at the court.
• In a trial a jury must decide between two hypotheses, the null hypothesis H0 is
o H0: The defendant is innocent.
• The alternative hypothesis HA is
o HA: The defendant is guilty.
• The jury does not know which hypothesis is true. They must make a decision on the
basis of evidence presented.
• In the language of statistics convicting the defendant is called
o rejecting the null hypothesis (the defendant is innocent) in favor of the
alternative hypothesis (the defendant is guilty).
• That is, the jury is saying that there is enough evidence to conclude that the
defendant is guilty (i.e., there is enough evidence to support the alternative
hypothesis).
• If the jury acquits it is stating that
o there is not enough evidence to support the alternative hypothesis.
• Notice that the jury is not saying that the defendant is innocent, only that there is
not enough evidence to support the alternative hypothesis. That is why we never say
that e aept the ull hypothesis that the defedat is ioet.
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Four possible outcomes from a hypothesis test
Type l and Type ll Errors
• Two possible errors can be made in any test.
o A Type I error occurs when we reject a true null hypothesis (i.e. reject H0
when H0 is true). In the criminal trial, a Type I error occurs when the jury
convicts an innocent person.
o A Type II error ours he e dot rejet a false ull hypothesis i.e. do ot
reject H0 when H0 is false). In a criminal trial, a Type II error occurs when a
guilty defendant is acquitted.
• The probability of a Type I error is denoted as (Greek letter alpha). The probability
of a Type II error is (Greek letter beta).
o P (making Type I error) =
o P (making Type II error) =
• is called the level of significance.
• The two probabilities are inversely related. Decreasing one increases the other.
In a criminal trial
• In our judicial system, Type I errors are regarded as more serious. We try to avoid
convicting innocent people (think about capital punishment!). Therefore, we are
more willing to acquit a guilty person.
• We arrange to make small by requiring the prosecution to prove its case and
istrutig the jury to fid the defedat guilty oly if there is eidee eyod a
reasoale dout.
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