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Embed code for: 7.1 Introducing Hypothesis Testing
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S1 Chapter 7: Hypothesis Testing Introducing Hypothesis Testing
Mr J Wooler firstname.lastname@example.org
In many real-life situations, people would like to “prove” things.
Medical scientists need to “prove” that a drug is safe and effective before it can be used in treatment.
Lawyers try to “prove” that a defendant is guilty or innocent.
In most cases like these, it is impossible to prove the case in the same way that it is possible to prove that Pythagoras’ theorem is true, for example.
In many cases, statistics are used as evidence to support a theory, particularly in drug testing.
It is important to realise that statistics can never prove anything.
Even if every patient given a particular drug during a clinical trial was cured, this does not prove that the drug caused them all to get better.
However, if the sample of patients was large enough, then this would be judged sufficient evidence that the drug works.
Even if some of the patients did not improve, the evidence could still be strong enough to support the hypothesis that the drug was effective.
This chapter is about using statistics to decide whether it is reasonable to accept a particular hypothesis or not.
In this first section, you will look at the process of setting up and carrying out a hypothesis test.
Using Binomial Tables
Firstly, some notes on using the Binomial tables.
Many of the questions will require you to be confident with using these tables.
Binomial tables give when for various values of and .
However, if the probability is not given in the table or the sample size is over 20 you must do the calculations.
Find the table for in your own set of tables (or there is one on page 174 of the textbook).
As we go through Example 1, check that you get the same probabilities when you look at the tables.
Look at the column for . Notice that all the cumulative frequencies from 10 to 19 are very close to 1 and to four decimal places are all rounded to 1.
Setting up a hypothesis test
In statistical work you often wish to find out if something that occurs is within the normal range of expectations or is an unusual occurrence.
Look at these three examples.
These three examples will help you understand how hypothesis testing works.
The ideal hypothesis test involves in this order:
Establish the null and alternative hypotheses
Decide on the significance level
Collect suitable data using a random sampling procedure that ensures the items are independent.
Conduct a test, doing the necessary calculations.
Interpret the results in terms of the original claim.
There are lots of situations where we cannot carry out a test as rigorously as this. In all the examples 2, 3 and 4 the data has already been collected before the hypotheses are set up.
However, it is important in S1 questions to read what you are investigating, rather than looking at the data to set up the hypothesis test.
It is important that in examples like this one you find the probability of obtaining the trial result or a more extreme result, in this case rather than .
This is a key concept in this chapter: when conducting a hypothesis test look for a region of probabilities.
Examples using larger samples
In examples 3 and 4 we calculated tail probabilities, i.e. a region.
Note, in example 2 we were calculatingwhich of course is the same as .
So although we may be surprised with the number of times the cricket captain guessed wrong, the number of questions the student got right and the number of students who passed the exam in examples 2, 3 and 4 respectively, the results turned out to be not significant: in all three cases the null hypothesis was accepted.
However, in all these cases the sample sizes were relatively small. What happens if we investigate results from larger samples?
Notice that the outcomes in examples 5, 6 and 7 are now all significant.
As sample sizes get larger, the conclusion may change even if the ratio of success stays the same.
The chance of guessing 10 or more questions correct out of 20 is much smaller than the chance of guessing 5 or more questions correct out of 10.
Notice from the examples that the conclusion should always be given in terms of the problem.
First state whether H0 is to be accepted or rejected, then make a statement beginning “there is evidence to suggest that …” or “there is not sufficient evidence to suggest that …”.
You should NOT write “this proves that ….” or “so the claim is right”. You are not proving anything, only considering evidence.
Using the Binomial Distribution
Pages 167 – 174
There is a lot of terminology and notation to learn, so work carefully and use the examples in the textbook and your notes to help
Questions 1, 4, 5, 7 & 8
Hypothesis testing gets easier with practice!
Exam Style Question
support the hypothesis that the drug was effective.