Hypothesis Testing for Correlation
You should be familiar with using a hypothesis test to determine bias within probability problems. It is also possible to use a hypothesis test to determine whether a given product moment correlation coefficient calculated from a sample could be representative of the same relationship existing within the whole population. For full information on hypothesis testing, see the revision notes from section 5.1.1 Hypothesis Testing
Why use a hypothesis test?
 In most cases it is too difficult to get the value of the PMCC for a whole population
 This would involve having data on each individual within the whole population
 It is very rare that a statistician would have the time or resources to collect all of that data
 The PMCC for the whole population can instead be estimated using information from a sample taken from the population
 The PMCC for a whole population is denoted (pronounced rho)
 The PMCC for a sample taken from the population is denoted r
 A hypothesis test would be conducted using the value of to r determine whether the population can be said to have positive, negative or zero correlation
How is a hypothesis test for correlation carried out?
 Most of the time the hypothesis test will be carried out by using a critical value
 You won't be expected to calculate pvalues but you might be given a pvalue
 Step 1. Write the null and alternative hypotheses clearly
 The hypothesis test could either be a onetailed test or a twotailed test
 The null hypothesis will always be
 The alternative hypothesis will depend on if it is a onetailed or twotailed test
 A onetailed test would test to see if the population PMCC, ρ, is either positive or negative
 The alternative hypothesis, H_{1} will be or
 A twotailed test would test to see if the population PMCC, ρ , is not equal to zero (meaning there is some form of linear correlation)
 The alternative hypothesis, H_{1} will be
 A onetailed test would test to see if the population PMCC, ρ, is either positive or negative
 Step 2. Either: Compare the value of r calculated from the sample with the critical value
 You will be given either a critical value or a p  value in the question
 You may be asked to find your own critical value from a table of values given in the question
 The table will be clear and easy to use, make sure you choose the correct level of significance for your test
 The table could have options for both a onetailed test and a twotailed test
 If you are given the critical value you should compare it with the sample PMCC, r
 If r is in the critical region the test is significant and the null hypothesis should be rejected


 It will be in the critical region if


 If r is not in the critical region the null hypothesis should be accepted and the alternative hypothesis should be rejected
 You will be given either a critical value or a p  value in the question
Or: Compare the p  value with the significance level

 You will not be expected to calculate the p  value, it will be given in the question
 If the p  value is less than the significance level the test is significant and the null hypothesis should be rejected
 If the p  value is greater than the significance level the null hypothesis should be accepted and the alternative hypothesis should be rejected
 You will not be expected to calculate the p  value, it will be given in the question
 Step 3. Write a conclusion in context
 Use the wording in the question to help you write your conclusion
 If rejecting the null hypothesis your conclusion should state that there is evidence to accept the context of the alternative hypothesis at the level of significance of the test only
 If accepting the null hypothesis your conclusion should state that there is not enough evidence to accept the context of the alternative hypothesis at the level of significance of the test only
Worked Example
A student believes that there is a positive correlation between the number of hours spent studying for a test and the percentage scored on it.
The student takes a random sample of 10 of his friends and records the amount of revision they did and percentage they score in the test.
The student calculates the product moment correlation coefficient for these data as .
Given that the critical value for this test is 0.5494, carry out a hypothesis test at the 5% level of significance to test whether the student’s claim is justified.
Exam Tip
 Make sure you read the question carefully to determine whether the test you are carrying out is for a onetailed or a twotailed test and use the level of significance accordingly. Be careful when comparing negative values of r with a negative critical value, it is easy to make an error with negative numbers when in an exam situation.