STAT1008 Study Guide - Final Guide: Null Hypothesis, Statistical Hypothesis Testing, Statistic
4.1 Hypothesis Tests
• Statistical test - uses data from a sample to assess a claim about a population.
• Statistical tests are framed formally in terms of two competing hypotheses:
• Null hypothesis (H0) - claim that there is no effect or no difference.
• Alternative hypothesis (Ha) - claim for which we seek significant evidence.
• The alternative hypothesis is established by observing evidence (data) that contradicts the null
hypothesis and supports the alternative hypothesis.
• H0 usually includes =
• Ha usually icludes >, < or ≠.
• The inequality of Ha depends on the question.
Statistical Significance
• When results as extreme as the observed sample statistic are unlikely to occur by random
chance alone (assuming the null hypothesis is true), we say the sample results are statistically
significant.
• If our sample is statistically significant, we have convincing evidence against H0 and in favour of
Ha.
• If our sample is not statistically significant, our test is inconclusive.
4.2 Measuring Evidence with P-Values
• To see if a statistic provides evidence against H0, we need to see what kind of sample statistics
we would observe, just by random chance if H0 was true.
• Randomisation distribution - a collection of statistics from samples simulated assuming the
null hypothesis is true.
• This makes it straightforward to assess how extreme the observed statistic is.
• P-value - the chance of obtaining a sample statistic as extreme (or more extreme) than the
observed sample statistic, if the null hypothesis is true.
• The p-value can be calculated as the proportion of statistics in a randomisation distribution
that are as extreme (or more extreme) than the observed sample statistic.
• The smaller the p-value, the stronger the statistical evidence is against the null hypothesis and
in favour of the alternative.
• What kinds of statistics would we get, just be random chance, if the null hypothesis were true?
(randomisation distribution)
• What proportion of these statistics are as extreme as our original sample statistic? (p-value)
Alternative hypothesis
• A one-sided alternative (we seek evidence in just one direction from the null value) contains
either > or <.
• A two-sided alternative (we are only looking to see if there is a difference without specifying
in advance in which direction) contains ≠.
• For a two-sided alternative, the p-value is twice the proportion in the smallest tail or double
the one-tail p-value.
• The smaller the p-value, the stronger the evidence against H0.
Estimating a P-value from a Randomisation Distribution
• For a one-tailed alternative - find the proportion of randomisation samples that equal or
exceed the original statistic in the direction (tail) indicated by the alternative hypothesis.
• For a two-tailed alternative - find the proportion of randomisation samples in the smaller tail
or beyond the original statistic and then double the proportion to account for the other detail.
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Document Summary
Statistical significance: when results as extreme as the observed sample statistic are unlikely to occur by random chance alone (assuming the null hypothesis is true), we say the sample results are statistically significant. If our sample is statistically significant, we have convincing evidence against h0 and in favour of. If our sample is not statistically significant, our test is inconclusive. For a two-sided alternative, the p-value is twice the proportion in the smallest tail or double the one-tail p-value: the smaller the p-value, the stronger the evidence against h0. For a one-tailed alternative - find the proportion of randomisation samples that equal or exceed the original statistic in the direction (tail) indicated by the alternative hypothesis. , is the threshold below which the p-value is deemed small enough to reject the null hypothesis: reject h0 if the p-value < . If the p-value is less than hypothesis in favour of the alternative.
