Hypothesis testing

STAT1003 – Statistical Techniques

Dr. Emi Tanaka

Australian National University

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Statistical inference

Estimation (last topic)

Draw inferences about a population by estimating population parameters from a sample (point and interval estimators).

Hypothesis testing (this topic)

Draw inferences about a population by making a claim or hypothesis about a population parameter and testing whether the hypothesis is supported by the sample.



The judicial system

  • In a criminal court case, the defendant is either:
    1. guilty of the alleged crime or
    2. not guilty of the alleged crime.
  • The prosecutor and the defence lawyer present their evidence with assumptions to the jury and/or judge during the trial.
  • The jury has to decide whether to convict the defendant or not based on the evidence.

The “statistical” judicial system

  • In a “statistical” court case, there are two competing hypotheses:
    1. The null hypothesis (\(H_0\)) or
    2. The alternative hypothesis (\(H_A\) or \(H_1\)).
  • A sample of data is collected and a test statistic under assumptions provides evidence to test the \(H_0\).
  • Based on the strength of evidence, we decide whether to reject the \(H_0\) or not.

Two types of errors can occur in this process:

  1. Type I error: rejecting the \(H_0\) when it is actually true.
  2. Type II error: failing to reject the \(H_0\) when the \(H_A\) is actually true.

Null hypothesis \(H_0\) and alternative hypothesis \(H_A\)

  • The null hypothesis, denoted as \(H_0\), is a statement about a population parameter (e.g. \(\theta = \theta_0\)) that we want to test.
  • \(H_0\) often represents a “status quo” or “no effect” scenario.
  • \(H_0\) is assumed to be true until evidence suggests otherwise.
  • The alternative hypothesis, denoted as \(H_A\) (or \(H_1\)), is a contrary statement to \(H_0\) that we want to provide evidence for.
  • The alternative hypothesis can be:
    • two-sided: \(H_A: \theta \neq \theta_0\) (two-tail test), or
    • one-sided: \(H_A: \theta > \theta_0\) (upper-tail test) or \(H_A: \theta < \theta_0\) (lower-tail test).

State the hypotheses

Determine the null and alternative hypotheses based on the research question.

The packaging on a light bulb states that the bulb will last 500 hours under normal use. A consumer advocate would like to know if the mean lifetime of a bulb is less than 500 hours.

According to an Australian Bureau of Statistics survey conducted in 2010, 77% of Australian adults were mostly or extremely satisfied with their life as a whole. A researcher wonders if the percentage of satisfied Australians is the same today.

A sports dietician would like to see whether the intake of caffeine prior to a race can improve the performance of ultra marathon runners.