Further Statistics 2: Chi-squared testsAQA A-Level Further Maths: Topic test

1. 1
   A college librarian asks 60 Year 12 students and 60 Year 13 students which revision method they prefer most. Of the Year 12 students, 30 prefer flashcards, 20 prefer past papers and 10 prefer video lessons. Of the Year 13 students, 20 prefer flashcards, 25 prefer past papers and 15 prefer video lessons. A chi-squared test is to be carried out to investigate whether there is an association between year group and preferred revision method.

   (a)

   How many degrees of freedom does the test have?

   [1 mark]

   • A$1$
   • B$3$
   • C$2$
   • D$5$

   (b)

   Find the expected frequency for Year 13 students who prefer video lessons.

   [1 mark]

   • A$12.5$
   • B$15$
   • C$25$
   • D$10$

   (c)

   Calculate the value of the test statistic.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A shop records how 300 customers pay: by card, contactless, cash or mobile app. The customers are classed in three age groups. A chi-squared test for association between age group and payment method is carried out using a $3\times4$ contingency table. The value of the test statistic is $13.8$.

   (a)

   How many degrees of freedom does the test have?

   [1 mark]

   • A$12$
   • B$11$
   • C$7$
   • D$6$

   (b)

   The $5\%$ critical value for the correct number of degrees of freedom is $12.592$. Which conclusion is correct?

   [1 mark]

   • ADo not reject $\mathrm{H}_0$: there is no evidence of an association between age group and payment method.
   • BReject $\mathrm{H}_0$: there is evidence of an association between age group and payment method.
   • CReject $\mathrm{H}_0$: age group causes customers to choose a particular payment method.
   • DReject $\mathrm{H}_0$: age group and payment method are independent.

   (c)

   One of the expected frequencies in the table is $3.6$. State what should be done before the test is carried out and why.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A vet records whether 50 dogs develop a skin reaction. Of the 25 dogs fed diet X, 10 develop a reaction. Of the 25 dogs fed diet Y, 3 develop a reaction. The vet carries out a chi-squared test at the $5\%$ significance level to investigate whether there is an association between diet and skin reaction. The $5\%$ critical value for $1$ degree of freedom is $3.841$.

   (a)

   Show that the expected frequency for dogs fed diet X that develop a reaction is $6.5$. Write down the number of degrees of freedom and explain why Yates' correction should be used.

   [3 marks]

   (b)

   Use Yates' correction to carry out the test. State your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two surveys are described. Survey 1: A university records the accommodation of 150 students (50 in halls, 50 in private rented housing and 50 living at home) and their attendance (high, medium or low). In halls, 28 have high attendance, 16 medium and 6 low. In rented housing, 18 have high, 24 medium and 8 low. Living at home, 14 have high, 20 medium and 16 low. Survey 2: A school asks 50 Year 10 and 40 Year 11 pupils for their favourite sport. Year 10: 22 football, 15 netball, 9 tennis, 4 athletics. Year 11: 18 football, 5 netball, 13 tennis, 4 athletics.

   (a)

   For Survey 1, carry out a chi-squared test at the $5\%$ significance level to investigate whether there is an association between accommodation and attendance. Identify the main sources of association. The $5\%$ critical value for $4$ degrees of freedom is $9.488$.

   [6 marks]

   (b)

   For Survey 2, explain why some of the data must be combined before a chi-squared test is carried out, and then test at the $5\%$ level whether there is an association between year group and favourite sport. The $5\%$ critical value for $2$ degrees of freedom is $5.991$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   In a care home trial of a hand-washing scheme, 30 residents took part in the scheme and 30 did not. Of those who took part, 4 caught an infection. Of those who did not take part, 16 caught an infection. A chi-squared test for association between the scheme and infection is to be carried out at the $1\%$ significance level using a $2\times2$ contingency table. The $1\%$ critical value for $1$ degree of freedom is $6.635$.

   (a)

   Why should Yates' correction be used in this test?

   [1 mark]

   • ASome expected frequencies are less than $5$.
   • BThe sample size is larger than $50$.
   • CThe table is $2\times2$, so there is one degree of freedom and the uncorrected statistic tends to be too large.
   • DThe two groups have equal sizes.

   (b)

   Find the value of the test statistic using Yates' correction.

   [1 mark]

   • A$10.8$
   • B$10.65$
   • C$6.05$
   • D$9.075$

   (c)

   State the conclusion of the test in context.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A survey of 240 shoppers, 60 at each of four shopping centres A, B, C and D, records the day each shopper prefers to visit: a weekday, Saturday or Sunday. In total, 80 shoppers prefer a weekday, 100 prefer Saturday and 60 prefer Sunday. At centre B, 10 shoppers prefer a weekday, 40 prefer Saturday and 10 prefer Sunday. A chi-squared test is carried out to investigate whether there is an association between shopping centre and preferred day.

   (a)

   How many degrees of freedom does the test have?

   [1 mark]

   • A$6$
   • B$12$
   • C$11$
   • D$5$

   (b)

   Find the expected frequency for shoppers at centre B who prefer Sunday.

   [1 mark]

   • A$10$
   • B$15$
   • C$25$
   • D$20$

   (c)

   Calculate the contribution to the test statistic from centre B shoppers who prefer Saturday, and state what it shows about the source of association.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   An airline records the punctuality of 100 flights from each of two airports, P and Q. From airport P, 62 flights were on time, 26 were delayed by less than an hour and 12 were delayed by an hour or more. From airport Q, 74 flights were on time, 16 were delayed by less than an hour and 10 were delayed by an hour or more. A chi-squared test at the $5\%$ significance level is used to investigate whether there is an association between airport and punctuality. The $5\%$ critical value for $2$ degrees of freedom is $5.991$.

   (a)

   Calculate the value of the test statistic.

   [3 marks]

   (b)

   Carry out the test, stating your hypotheses, the degrees of freedom and your conclusion in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two surveys are described. Survey 1: A zoo asks 200 visitors whether they bought a guidebook. Of 60 children, 12 bought one. Of 100 adults, 40 bought one. Of 40 seniors, 18 bought one. Survey 2: A bakery records whether customers made a repeat purchase within a week. Of 30 customers with a loyalty card, 24 made a repeat purchase. Of 30 customers without one, 15 did.

   (a)

   For Survey 1, carry out a chi-squared test at the $5\%$ significance level to investigate whether age group and buying a guidebook are associated. Identify the main source of association. The $5\%$ critical value for $2$ degrees of freedom is $5.991$.

   [6 marks]

   (b)

   For Survey 2, use Yates' correction to test at the $5\%$ significance level whether there is an association between holding a loyalty card and making a repeat purchase. Explain why the correction is used. The $5\%$ critical value for $1$ degree of freedom is $3.841$.

   [6 marks]

   Total for question 8: 12 marks

End of questions