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 |
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is a two-sided alternative |
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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 |
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the alternative hypothesis is written: H1: µ =/= µo |
One-tailed test |
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one-sided alternative |
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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 |
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E.g. the alternative hypothesis is written as: H1: µ > µo |
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Decision-making process steps
1. |
Collecting the data |
2. |
Summarising the data |
3. |
Setting up a hypothesis (i.e. a claim or theory), which is to be tested |
4. |
Calculating the probability of obtaining a sample such as the one we have if the hypothesis is true |
5. |
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 |
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the size (n) of the sample |
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the mean of the sample |
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the standard deviation (s) of the sample |
Other information of interest might include: |
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Does the population have a normal distribution? |
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Is the population’s standard deviation known? |
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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) |
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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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