Contingency tables and the chi-squared testAQA A-Level Further Maths: Subtopic test

1. 1
   A survey of 150 employees records how they travel to work and whether they live in town. Of the 40 who walk, 30 live in town. Of the 60 who take the bus, 25 live in town. Of the 50 who drive, 15 live in town. A chi-squared test for association is to be carried out.

   (a)

   Assuming no association, find the expected number of employees who drive and live in town.

   [1 mark]

   • A$25.0$
   • B$26.7$
   • C$23.3$
   • D$35.0$

   (b)

   Find the contribution to the test statistic from the employees who walk and live in town.

   [1 mark]

   • A$4.28$
   • B$11.3$
   • C$128$
   • D$6.88$

   (c)

   State the null hypothesis and the number of degrees of freedom for the test.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Hospital records for 200 patients from three wards, A, B and C, give each patient's recovery outcome. Ward A has 70 patients: 48 recovered fully, 20 partially and 2 not at all. Ward B has 60 patients: 35 fully, 22 partially and 3 not at all. Ward C has 70 patients: 27 fully, 36 partially and 7 not at all.

   (a)

   Assuming recovery is independent of ward, find the expected number of patients in ward B who recovered partially.

   [1 mark]

   • A$23.4$
   • B$22.0$
   • C$26.0$
   • D$20.0$

   (b)

   How many degrees of freedom does the test statistic for this table have?

   [1 mark]

   • A$8$
   • B$4$
   • C$9$
   • D$6$

   (c)

   Explain why two of the outcome categories must be combined before the test is carried out, and state the degrees of freedom after doing so.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cafe owner records the drink chosen by 120 customers: 60 in the morning and 60 in the afternoon. In the morning, 18 chose tea, 30 chose coffee and 12 chose juice. In the afternoon, 24 chose tea, 14 chose coffee and 22 chose juice.

   (a)

   Find the expected frequency for each cell, assuming the drink is independent of the time of day, and calculate the value of the test statistic $\sum\frac{(O-E)^2}{E}$.

   [3 marks]

   (b)

   Using the test statistic $9.62$, test at the $5\%$ significance level whether the drink chosen is associated with the time of day.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A health authority surveys 180 patients, 60 at each of three clinics, X, Y and Z, about their satisfaction. At X, 40 were satisfied, 15 neutral and 5 dissatisfied. At Y, 30 were satisfied, 20 neutral and 10 dissatisfied. At Z, 20 were satisfied, 20 neutral and 20 dissatisfied.

   (a)

   Carry out a chi-squared test, at the $5\%$ significance level, to investigate whether satisfaction is associated with the clinic.

   [6 marks]

   (b)

   The test in part (a) shows an association. Use the contributions of the individual cells to the test statistic to identify the sources of the association, and state a conclusion in context.

   [6 marks]

   Total for question 4: 12 marks

End of questions