Hypothesis tests for correlationAQA A-Level Maths: Subtopic test

1. 1
   A teacher records the number of hours $h$ that each of $12$ students spent revising and the student's test score $s$. The product moment correlation coefficient for the sample is $r=0.62$. The teacher tests, at the $5\%$ significance level, whether there is positive correlation between hours revising and test score in the population. For a sample of size $12$, the one-tailed $5\%$ critical value of the product moment correlation coefficient is $0.4973$.

   (a)

   Which pair of hypotheses is correct for the teacher's test?

   [1 mark]

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

   (b)

   Which conclusion is correct?

   [1 mark]

   • ADo not reject $H_0$: there is no correlation between hours revising and test score
   • BReject $H_0$: more hours of revising cause higher test scores
   • CReject $H_0$: there is sufficient evidence of positive correlation between hours revising and test score
   • DDo not reject $H_0$, because $0.62$ is greater than $0.4973$

   (c)

   A student says, 'The test shows that revising for more hours causes higher scores.' Explain why this conclusion is not justified.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A researcher takes a random sample of $10$ adults and finds that the product moment correlation coefficient between hours of screen time per day and hours of sleep is $r=-0.47$. She tests, at the $5\%$ significance level, whether there is negative correlation in the population. For a sample of size $10$, the one-tailed $5\%$ critical value is $0.5494$.

   (a)

   The critical region for the sample correlation coefficient $r$ is

   [1 mark]

   • A$r\ge0.5494$
   • B$r\le-0.5494$
   • C$r\le-0.6319$
   • D$-0.5494\le r\le0.5494$

   (b)

   Which conclusion is correct for $r=-0.47$?

   [1 mark]

   • AReject $H_0$: there is strong evidence of negative correlation
   • BThere is no correlation between screen time and sleep
   • C$\lvert r\rvert<0.5494$, so the population correlation is zero
   • DDo not reject $H_0$: there is insufficient evidence at the $5\%$ level of negative correlation between screen time and sleep

   (c)

   Software gives a $p$-value of $0.085$ for this test. Use the $p$-value to state the conclusion of the test at the $10\%$ significance level.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A random sample of $20$ students has product moment correlation coefficient $r=0.41$ between hours of sleep and score in a memory test. For a sample of size $20$, the critical values are: one-tailed $5\%$, $0.3783$; two-tailed $5\%$, $0.4438$.

   (a)

   Test, at the $5\%$ significance level, whether there is correlation (positive or negative) between hours of sleep and memory test score.

   [3 marks]

   (b)

   Another analyst tests $H_0:\rho=0$ against $H_1:\rho>0$ at the $5\%$ level, using the same data, and concludes that there is evidence of positive correlation. Explain why the two conclusions are not inconsistent, and say when a one-tailed test is justified.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A geographer collects data on the distance $d$ km of each of $15$ villages from a large city and the number of residents $p$ in each village. All the distances are between $3$ km and $30$ km. The product moment correlation coefficient is $r=-0.52$. For a sample of size $15$, the two-tailed critical values are: $5\%$ significance level, $0.5140$; $1\%$ significance level, $0.6411$.

   (a)

   (i) Write down suitable hypotheses for a test of whether there is correlation between distance from the city and population.
   (ii) Carry out the test at the $5\%$ significance level.
   (iii) Carry out the test at the $1\%$ significance level, and comment on the strength of the evidence.

   [6 marks]

   (b)

   The geographer concludes: 'The correlation shows that living further from the city makes villages smaller, so I can use the data to predict the population of a village $60$ km from the city.' Evaluate her conclusion.

   [6 marks]

   Total for question 4: 12 marks

End of questions