week 5.pdf - 21/11/2018 Hypothesis testing; Sign Test Week...

21/11/2018
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Hypothesis testing;
Sign Test
Week 5
1
Formal inference: hypothesis testing
• Aim: to make objective inference from something that is just a sample
to a whole population
• Method: take a random sample from a particular population, then use
this sample to infer/deduce information about the whole population
• We will use formal statistical tests such as the Sign Test, Z ‐ test, t ‐ test,
chi ‐ squared test, ANOVA and linear regression – as well as confidence
intervals – for the rest of this course
2
FAQs
• So how can we use a sample to ‘prove’ what we suspect is true?
• Using a sample, we cannot prove beyond all doubt that a suspicion or belief
about the population is true or false!
• Why do we need probability for Statistics?
• Probability theory deals with the analysis of random phenomena. We use
probability theory to find whether or not our result is a significant finding in a
statistical sense (i.e. statistically significant) and if so, at what level our result
is significant
• We use Statistics (and probability in particular) to “quantify our uncertainty”
about our findings
[we could only do this if we sampled
every single member of the population,
as in a census]
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21/11/2018
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What would surprise you?
Would you draw any conclusions from these sample statistics?
• In a Statistics lecturer’s ante ‐ natal class, 7 out of the 8 couples gave
birth to a boy
• From a sample of 15 CEOs, 3 are female
• We toss a coin 100 times and it lands on heads 76 times
• The mean height of a random sample of 20 university students is 1.9m
• There are proportionally far more people with a high level of
understanding of te reo in the younger age groups
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Hypothesis testing
A fundamental component of Statistical Analysis
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New terms
• Hypothesis
 Null hypothesis
 Alternative hypothesis
• Significance level
• Test statistic
• p ‐ value
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21/11/2018
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Hypothesis testing – the main idea
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Hypothesis testing – the process
• We start with a belief or suspicion about a population – this belief is our
hypothesis. For example
• We collect information from a random sample, which is taken from the
population that the hypothesis is about
• Use the sample results and statistical procedures to test whether the
suspicion we have is likely to be true for the whole population
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Construct two opposing statements about your belief
(these may be written in symbols):
 "ܪ
଴
" Null hypothesis
(sceptic’s point of view, there is no change/difference)
e.g. there is no bias in the coin;
there is no change in median cortex thickness;
the median survival time does not differ between drugs
 "ܪ
ଵ
" Alternative hypothesis
(the change you suspect, ‘research hypothesis’)
e.g. the coin is biased;
there is a change in cortex thickness;
the average survival time is longer with drug B than with drug A
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21/11/2018
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 Null hypothesis ܪ
଴
an equality stated in terms of a population parameter
a statement of the value of the population parameter when there is no change
e.g. ܪ
଴
: population median (or mean) ൌ a particular number
 Alternative hypothesis ܪ
ଵ