Hypothesis Testing for Correlation (DP IB Maths: AI HL)

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Dan

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Dan

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Hypothesis Testing for Correlation

What is a hypothesis test for correlation?

  • You can use a t-test to test whether there is linear correlation between two normally distributed variables
    • If specifically testing for positive (or negative) linear correlation then a one-tailed test is used
    • If testing for any linear correlation then a two-tailed test is used
  • A sample will be taken and the raw data will be given
    • You might be asked to calculate the PMCC (Pearson's product-moment correlation coefficient)

What are the steps for a hypothesis test for correlation?

  • STEP 1: Write the hypotheses
    • H0 : ρ = 0
      • Clearly state that ρ represents population correlation coefficient between the two variables
      • In words this means there is no correlation
    • H1 : ρ < 0, H1 : ρ > 0 or H1 : ρ ≠ 0
  • STEP 2: Calculate the p-value or the PMCC
    • Choose a t-test for linear regression
    • Enter the data as two lists into GDC
  • STEP 3: Decide whether there is evidence to reject the null hypothesis
    • If the p-value < significance level then reject H0
    • If the absolute value of the PMCC is bigger than the absolute value of the critical value then reject H0
      • If you are expected to use the PMCC you will be given the critical value in the exam
  • STEP 4: Write your conclusion
    • If you reject H0­ then there is evidence to suggest that...
      • There is a negative linear correlation between the two variables (for H1 : ρ < 0)
      • There is a positive linear correlation between the two variables (for H1 : ρ > 0)
      • There is a linear correlation between the two variables (for H1 : ρ ≠ 0)
    • If you accept H­0 then there is insufficient evidence to reject the null hypothesis which suggests that...
      • There is not a negative linear correlation between the two variables (for H1 : ρ < 0)
      • There is not a positive linear correlation between the two variables (for H1 : ρ > 0)
      • There is not a linear correlation between the two variables (for H1 : ρ ≠ 0)

Worked example

Jessica wants to test whether there is any linear correlation between the distance she runs in a day, d km, and the amount of sleep she has the night after her run, t hours. Over the period of a month she takes a random sample of 9 days, the results are recorded in the table. 

Distance (d km)

1.2

2.3

1.5

1.3

2.5

1.8

1.9

2.0

1.1

Sleep (t hours)

7.9

8.1

7.6

7.3

8.1

8.4

7.8

7.9

6.8

a)
Write down null and alternative hypotheses that Jessica can use for her test.

4-12-6-ib-ai-hl-hyp-test-for-correlation-a-we-solution

b)
Perform the hypothesis test for linear correlation using a 5% significance level. Clearly state your conclusion.

4-12-6-ib-ai-hl-hyp-test-for-correlation-b-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.