Testing for zero correlationEdexcel A-Level Further Maths: Subtopic test

1. 1
   For a random sample of 10 adults, the product moment correlation coefficient between weekly hours of exercise and resting heart rate is $r=-0.60$. A test is to be carried out, at the 5% significance level, to see whether there is any correlation between the two variables in the population.

   (a)

   Which pair of hypotheses is correct for this test?

   [1 mark]

   • A$H_0: r=0,\ H_1: r\ne0$
   • B$H_0: \rho=0,\ H_1: \rho<0$
   • C$H_0: \rho=0,\ H_1: \rho\ne0$
   • D$H_0: \rho\ne0,\ H_1: \rho=0$

   (b)

   Using the critical value $0.6319$ from the table, what is the correct conclusion?

   [1 mark]

   • ADo not reject $H_0$: there is insufficient evidence of correlation at the 5% level
   • BReject $H_0$: there is evidence of negative correlation at the 5% level
   • CDo not reject $H_0$: this proves there is no correlation
   • DReject $H_0$ because $-0.60<0.6319$

   (c)

   The researcher now only wants to test for negative correlation, using $H_0:\rho=0$ and $H_1:\rho<0$ at the 5% significance level. Using the critical value $-0.5494$, carry out the test and state your conclusion in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A teacher ranks 8 students by their coursework mark and by their end-of-year examination mark. Spearman's rank correlation coefficient between the two sets of ranks is $r_s=0.69$. The teacher tests, at the 5% significance level, whether there is positive correlation between coursework and examination performance.

   (a)

   Which pair of hypotheses is correct for this test?

   [1 mark]

   • A$H_0: \rho_s=0,\ H_1: \rho_s\ne0$
   • B$H_0: \rho_s>0,\ H_1: \rho_s=0$
   • C$H_0: \rho_s=0.69,\ H_1: \rho_s>0.69$
   • D$H_0: \rho_s=0,\ H_1: \rho_s>0$

   (b)

   Which critical value from the table of Spearman's rank correlation coefficient should be used?

   [1 mark]

   • A$0.7381$
   • B$0.6429$
   • C$0.5636$
   • D$0.8333$

   (c)

   Using the critical value $0.6429$, complete the test and state your conclusion in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A farmer applies different masses of fertiliser to 12 randomly chosen plots of land. The product moment correlation coefficient between the mass of fertiliser and the crop yield is $r=0.65$.

   (a)

   Test, at the 5% significance level, whether there is positive correlation between the mass of fertiliser and the crop yield. Use the critical value $0.4973$ and state the hypotheses clearly.

   [3 marks]

   (b)

   The farmer's adviser wants stronger evidence and uses the 1% significance level, with the critical value $0.6581$. Carry out this test. State one assumption about the population that is needed, and how a scatter diagram could be used to check it.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student investigates whether there is correlation between the hours of study per week, $x$, and the score in a test, $y$, for 8 randomly chosen students. The data are: $x$: $5, 12, 8, 20, 15, 3, 10, 18$ and $y$: $60, 52, 49, 66, 80, 45, 63, 71$, listed in the same student order.

   (a)

   Calculate Spearman's rank correlation coefficient for these data.

   [6 marks]

   (b)

   (i) Test, at the 5% significance level, whether there is correlation between hours of study and test score, using the critical value $0.7381$ and your answer to (a) (if you did not obtain an answer to (a), use $r_s=0.786$).
   (ii) The student also finds that the product moment correlation coefficient is $0.703$, and the critical value for a two-tailed 5% test is $0.7067$. Explain what this suggests about the conclusion in (i).

   [6 marks]

   Total for question 4: 12 marks

End of questions