Hypothesis test for correlationEdexcel A-Level Maths: Subtopic test

1. 1
   A student records, on 20 randomly chosen days, the daily maximum temperature and the number of ice creams sold at a kiosk. The product moment correlation coefficient is $r=0.52$. The student tests for correlation at the 5% significance level. For a two-tailed test with sample size 20, the critical value at the 5% level is $0.4438$.

   (a)

   Which pair of hypotheses should the student use?

   [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 describes the critical region for this test?

   [1 mark]

   • A$r>0.4438$ only
   • B$-0.4438<r<0.4438$
   • C$r<-0.4438$ only
   • D$r<-0.4438$ or $r>0.4438$

   (c)

   Carry out the test and state your conclusion in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A teacher records, for 12 students, the time spent on screens each night and the score in a memory test. The product moment correlation coefficient is $r=-0.62$. The teacher believes that more screen time is associated with lower scores, and tests this at the 5% significance level. For a one-tailed test with sample size 12, the critical value at the 5% level is $0.4973$.

   (a)

   Which alternative hypothesis should the teacher use?

   [1 mark]

   • A$H_1:\rho\neq0$
   • B$H_1:\rho>0$
   • C$H_1:\rho<0$
   • D$H_1:r<0$

   (b)

   What is the correct conclusion?

   [1 mark]

   • AReject $H_0$ in favour of $H_1$, because $|-0.62|>0.4973$
   • BDo not reject $H_0$, because $-0.62$ is less than $0.4973$
   • CDo not reject $H_0$, because $r$ is negative
   • DReject $H_0$ in favour of $H_1$, because $0.62>0.05$

   (c)

   A student says: 'The result proves that spending more time on screens makes scores worse.' Comment on this statement.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A farmer records the amount of fertiliser, $x$ kg per hectare, and the yield, $y$ tonnes per hectare, on 8 fields. The data are $(20,4.1)$, $(30,4.9)$, $(40,4.4)$, $(50,5.8)$, $(60,5.1)$, $(70,5.6)$, $(80,5.9)$, $(90,5.3)$. For a one-tailed test with sample size 8, the critical value is $0.6215$ at the 5% level and $0.7887$ at the 1% level.

   (a)

   Use your calculator to find the product moment correlation coefficient $r$, and state suitable hypotheses for a test of whether more fertiliser is associated with a greater yield.

   [3 marks]

   (b)

   Test at the 5% level and then at the 1% level, stating your conclusion each time in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sports scientist measures, for a random sample of 30 athletes, the weekly hours of strength training and the height of each athlete's best jump. The product moment correlation coefficient is $r=0.41$. For a two-tailed test with sample size 30, the critical value is $0.3610$ at the 5% level and $0.4629$ at the 1% level.

   (a)

   (i) Write down suitable hypotheses for a two-tailed test.
   (ii) Test at the 5% level and state your conclusion in context.
   (iii) State the conclusion of the test at the 1% level.

   [6 marks]

   (b)

   The scientist's coach says: 'This proves that strength training causes higher jumps, and the relationship is strong.' Evaluate the coach's claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions