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Hypothesis Testing Cheat Sheet by

Statis­tical inference

Draw conclu­sions from a set of data
Put a probab­ility on whether a conclusion is correct ‘beyond reasonable doubt’
The major question to answer is whether a difference between samples, or between a sample and a popula­tion, has occurred simply as a result of natural variation or because of a real difference between the two

Two-tailed or one-tailed

The altern­ative hypothesis may be classified as two-tailed or one-tailed
Two-­tailed test
is a two-sided altern­ative
we do the test with no precon­ceived notion that the true value of μ is either above or below the hypoth­esised value of μ 0
the altern­ative hypothesis is written: H1: µ =/= µo
One-­tailed test
one-sided altern­ative
do the test with a strong conviction that, if H0 is not true, it is clear that m is either grater than µ0 or less than µ0
E.g. the altern­ative hypothesis is written as: H1: µ > µo

Decisi­on-­making process steps

Collecting the data
Summar­ising the data
Setting up a hypothesis (i.e. a claim or theory), which is to be tested
Calcul­ating the probab­ility of obtaining a sample such as the one we have if the hypothesis is true
Either accepting or rejecting the hypothesis

Signif­icance level

After the approp­riate hypotheses have been formul­ated, we must decide upon the signif­icance level (or α -level) of the test
most common signif­icance level used is 0.05, commonly written as α = 0.05
A 5% signif­icance level says in effect that an event has occurred that occurs less than 5% of the time is considered unusual

One-sample z-test

Deals with the case of a single sample being chosen from a population and the question of whether that particular sample might be consistent with the rest of the population
Construct a test statistic according to a particular formula
Inform­ation required in calcul­ation
the size (n) of the sample
the mean of the sample
the standard deviation (s) of the sample
Other inform­ation of interest might include:
Does the population have a normal distri­bution?
Is the popula­tion’s standard deviation known?
Is the sample size (n) large? (25+)
There are different cases for the one-sample z-test statistic
Case I
the population has a normal distri­bution and
the population standard deviation, s, is known
Case II
the population has any distri­bution
the sample size, n, is large (i.e. at least 25), and
the value of population standard deviation is known
In both these cases we can use a z-test statistic formula (a)
Case III
the population has any distri­bution
the sample size, n, is large (i.e. at least 25), and
the value of population standard devation is unknown (however, since n is large, the value of population standard devation is approx­imated by the sample standard deviation, s)
In this case we can use a z-test statistic formula (b)

Set up your Hypothesis

Null Hypoth­esis
Part of formul­ation of an hypothesis
Statement that nothing unusual has occurred
The notation is Ho
Alte­rnative hypothesis
States that something unusual has occurred
The notation is H1 or HA
Toge­ther they may be written in the form: Ho: (state­ment) v. H1(alt­ern­ative statement)

Conclusion errors

Two possible errors in making a conclusion about a null hypothesis
Type I errors occur when you reject H0 (i.e. conclude that it is false) when H0 is really true.
Type II errors occur when you accept H0 (i.e. conclude that it is true) when H0 is really false.

z-test statistic formula (a)

z-test statistic formula (b)

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