2.5.2 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 p-values but you might be given a p-value
• Step 1. Write the null and alternative hypotheses clearly
  • The hypothesis test could either be a one-tailed test or a two-tailed test
  • The null hypothesis will always be 
  • The alternative hypothesis will depend on if it is a one-tailed or two-tailed test
    • A one-tailed test would test to see if the population PMCC, ρ, is either positive or negative
      • The alternative hypothesis, H₁ will be  or  
    • A two-tailed 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, H₁ will be  
• Step 2. Either: Compare the value of r calculated from the sample with the critical value
  • You will be given the critical value in the question 
  • 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

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
• 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