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

1. 1
   A teacher records the hours of revision and the test score of each of 10 randomly chosen students. The data may be assumed to come from a bivariate Normal distribution. The product moment correlation coefficient for the sample is $r=0.62$. The teacher tests at the 5% significance level whether there is any correlation between revision hours and score. For $n=10$ the two-tailed 5% critical value is 0.6319.

   (a)

   Which pair of hypotheses should the teacher use, where $\rho$ is the population product moment correlation coefficient?

   [1 mark]

   • A$H_0$: $\rho=0$, $H_1$: $\rho>0$
   • B$H_0$: $\rho=0$, $H_1$: $\rho\neq0$
   • C$H_0$: $r=0$, $H_1$: $r\neq0$
   • D$H_0$: $\rho\neq0$, $H_1$: $\rho=0$

   (b)

   Which conclusion is correct?

   [1 mark]

   • AReject $H_0$, because $0.62$ is greater than $0.05$
   • BReject $H_0$: there is evidence of correlation between revision hours and score
   • CThere is insufficient evidence to reject $H_0$, because $0.62<0.6319$: no evidence of correlation
   • D$H_0$ is true, so revision hours and score are not related

   (c)

   Suppose instead that the relationship between revision hours and score were known to be increasing but clearly curved. Name a more suitable measure of correlation and give a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A tutor believes that students who complete more practice papers achieve higher mock exam ranks. For a random sample of 8 students, the number of practice papers and the mock mark are each ranked with rank 1 for the highest, and $\sum d^2=26$. For $n=8$ the 5% critical value for Spearman's coefficient is 0.6429 for a one-tailed test and 0.7381 for a two-tailed test.

   (a)

   Which alternative hypothesis should the tutor use, where $\rho_s$ is the population rank correlation coefficient?

   [1 mark]

   • A$H_1$: $\rho_s>0$
   • B$H_1$: $\rho_s\neq0$
   • C$H_1$: $\rho_s<0$
   • D$H_1$: $r_s>0$

   (b)

   Calculate the sample value of Spearman's rank correlation coefficient.

   [1 mark]

   • A$0.310$
   • B$0.700$
   • C$-0.690$
   • D$0.690$

   (c)

   Carry out the test at the 5% significance level and state the conclusion in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cafe owner believes that daily sales of hot drinks fall as the outside temperature rises. Over 12 randomly chosen days the owner records the temperature and the number of hot drinks sold. The data may be assumed to come from a bivariate Normal distribution, and the sample product moment correlation coefficient is $r=-0.632$. For $n=12$ the one-tailed critical values for the product moment correlation coefficient are 0.4973 at the 5% level and 0.6581 at the 1% level.

   (a)

   State the hypotheses for the owner's test and the critical region at the 5% level.

   [3 marks]

   (b)

   Carry out the test at the 5% level, stating your conclusion in context, and say what the result would be at the 1% level.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school investigates whether weekly screen time and average nightly sleep are correlated. For ten randomly chosen students A to J, the weekly screen times in hours are 12, 25, 8, 30, 18, 5, 22, 15, 28 and 10, and the average nightly sleep in hours is 8.0, 6.5, 8.5, 6.0, 7.2, 8.8, 7.5, 6.9, 6.6 and 7.8 respectively. For $n=10$ the two-tailed 5% critical value for Spearman's coefficient is 0.6485.

   (a)

   Test at the 5% significance level whether there is correlation between screen time and sleep, using Spearman's rank correlation coefficient. State your conclusion in context.

   [6 marks]

   (b)

   A second school tests ten students in the same way and finds $r_s=-0.55$. Carry out the two-tailed test at the 5% level for this school and comment on the two results.

   [6 marks]

   Total for question 4: 12 marks

End of questions