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Hypothesis Testing
Hypothesis testing allows us to use a sample to decide between two statements made about a Population characteristic.
Population Characteristics are things like “ The mean of a population” or “ the proportion of the population who have a particular property”.
These two statements are called the Null Hypothesis and the Alternative Hypothesis.
Definitions
H 0 : The Null Hypothesis This is the hypothesis or claim that is initially assumed to be true.
HA : The Alternative Hypothesis This is the hypothesis or claim which we initially assume to be false but which we may decide to accept if there is sufficient evidence.
This procedure is familiar to us already from the legal system: “Innocent until proven guilty”.
The Null Hypothesis (Innocent) is only rejected in favour of the Alternative Hypothesis (Guilty) if
there is sufficient evidence of this “beyond reasonable doubt”.
So to summarise H 0 is the status quo and H (^) A is what we want to prove using the data we have collected.
H 0 and HA take the following form:
Null Hypothesis H 0 : Population Characteristic = Hypothesised Value
Alternative Hypothesis ~ three possibilities
Upper tailed test: HA : Population Characteristic > Hypothesised Value
Lower tailed test: HA : Population Characteristic < Hypothesised Value
Two tailed test: HA : Population Characteristic ≠ Hypothesised Value
NOTE: The same Hypothesised value must be used in the Alternative Hypothesis as in the Null Hypothesis
Definitions
- The probability of a Type I error is called the Level Of Significance of the Hypothesis Test and is denoted by α -alpha.
- The probability of a Type II error is denoted by β.
We choose α to be small ( .01, .05 or .1) but we cannot completely eliminate the probability of a Type I error, as we mentioned already α and β are related
The only way to reduce β without increasing α is to
increase the sample size.
Type I errors are generally considered the more serious so in our testing procedure we control the probability of these errors ( α ) and usually are
unaware of the probability of Type II errors ( β ).
Rejection Regions
One way of performing a Hypothesis test is to compute a rejection region. If we find that our test statistic (which we measure from our sample) is in this region then we reject our NULL Hypothesis.
The computation of the rejection region is mathematical and involves us using statistical tables like the Normal tables.
What is important is the idea that the rejection region is a region far away from our Null hypothesis. And that it is unlikely that we would observe a sample with a value of the test statistic (for example the sample mean or sample proportion) this far away from the Null Hypothesised value if that Null Hypothesis was true.
For example in a class of 100 people.
Suppose our Null Hypothesis is that the average age of the whole class is 31. Suppose we now observe a value of 20 in a sample we have randomly chosen. That’s very unlikely to have happened by chance if our Null Hypothesis was true. It’s much more likely that our Null Hypothesis is false. So we decide to reject our Null
Section P-Values
Introduction The conclusion of a Hypothesis test is dependent on the initial choice of α.
- Remember it was this Significance Level which determined the Rejection Region.
- Remember also that α = Probability of a Type I error.
In these tests we decided what value of α “we could live with” before we conducted the test.
Aware as you are that decreasing α increases β, you might choose a different value for α than I would.
Depending on these choices of α we may come to different conclusions about whether to reject the Null Hypothesis.
An alternative approach to Hypothesis testing is not to state any α at the outset but to wait until the Test Statistic has been calculated and then decide
whether to reject or not by comparison with α. This way all the calculations can be performed independent of any individual choice of α and at
the end each person can make a decision based on their choice of α.
How would we conduct such a test? Instead of computing a Rejection Region we calculate a P-Value.
IMPORTANT: Small P-values are equivalent to values of the test statistic far away from the NULL. Don’t worry about their computation.
Definition
The P-Value is the smallest level of significance at which H 0 can be rejected.
