1. A university required all Science students to study one language for one year.
A survey was carried out at the university amongst the 150 Science students. These students all studied one of either French, Spanish or Russian. The results of the survey are shown below.
French Spanish Russian
Female 9 29 12
Male 31 40 29
Ludmila decides to use the χ2 test at the 5 % level of significance to determine whether the choice of language is independent of gender.
(a) State Ludmila’s null hypothesis.
(1)
(b) Write down the number of degrees of freedom.
(1)
(c) Find the expected frequency for the females studying Spanish.
(2)
(d) Use your graphic display calculator to find the χ2 test statistic for this data.
(2)
(e) State whether Ludmila accepts the null hypothesis. Give a reason for your answer.
(2) (Total 8 marks)
2. Tony wants to carry out a χ2 test to determine whether or not a person’s choice of one of the three professions – engineering, medicine or law – is influenced by the person’s sex (gender).
(a) State the null hypothesis, H0, for this test.
(1)
(b) Write down the number of degrees of freedom.
(1)
Of the 400 people Tony interviewed, 220 were male and 180 were female. 80 of the people had chosen engineering as a profession.
(c) Calculate the expected number of female engineers.
(2)
Tony used a 5 % level of significance for his test and obtained a p-value of 0.0634 correct to 3 significant figures.
(d) State Tony’s conclusion to the test. Give a reason for this conclusion.
(2) (Total 6 marks)
3. A market researcher consulted males and females to determine whether the type of coffee they drink is associated with gender. The types of coffee are Cappuccino, Latte, Americano, Macchiato and Espresso. A χ2 test was conducted, at the 5 % significance level, and the χ2 value was found to be 8.73.
(a) Write down
(i) the null hypothesis;
(ii) the alternative hypothesis.
(2)
(b) Write down the number of degrees of freedom for this test.
(1)
(c) Write down the critical value for this test.
(1)
(d) State whether the type of coffee drunk is independent of gender. Give a reason for your answer.
(2) (Total 6 marks)
4. The same 100 students are also asked how many meals on average they have per day. The data collected is organized in the following table.
3 or fewer meals per day
4 or 5 meals per day
More than
5 meals per day Total
Male 15 25 15 55
Female 12 20 13 45
Total 27 45 28 100
A χ2 test is carried out at the 5 % level of significance. (a) Write down the null hypothesis, H0, for this test.
(1)
(b) Write down the number of degrees of freedom for this test.
(1)
(c) Write down the critical value for this test.
(1)
(d) Show that the expected number of females that have more than 5 meals per day is 13, correct to the nearest integer.
(2)
(e) Use your graphic display calculator to find the χ2calc for this data.
(2)
(f) Decide whether H0 must be accepted. Justify your answer.
(2) (Total 9 marks)
5. A questionnaire was given to all members of a school community to find out which drink was the most popular to have with breakfast. The results are given in the table below, classified by age.
Hot Chocolate Tea Coffee Milk
Children aged 12 years and less 55 10 1 34
Teenagers aged from 13 to 19 years 25 35 20 10
Adults aged 20 years and over 20 40 79 6
A χ2 test was conducted to decide whether the type of drink was independent of age. (a) Find the number of degrees of freedom for the χ2 test.
(2)
(b) Write down the null hypothesis for the χ2 test.
(1)
(c) Write down the critical value for the χ2 test at the 5 % significance level.
(1)
The χ2 test statistic is calculated to be 146 with a p-value of 6.62 × 10–29 (both numbers given correct to 3 significant figures).
(d) Write down the conclusion reached at the 5 % significance level. Give a clear reason for your answer.
(2) (Total 6 marks)
6. In a study on 100 students there seemed to be a difference between males and females in their choice of favourite car colour. The results are given in the table below.
A χ2 test was conducted.
(a) Write down the total number of male students.
(1)
(b) Show that the expected frequency for males, whose favourite car colour is blue, is 12.6.
(2)
The calculated value of χ2 is 1.367.
(c) (i) Write down the null hypothesis for this test.
(ii) Write down the number of degrees of freedom.
(iii) Write down the critical value of χ2 at the 5 % significance level.
(iv) Determine whether the null hypothesis should be accepted. Give a reason for your answer.
(5) (Total 8 marks)
Blue Red Green
Males 14 6 8
SOLUTIONS
1. (a) H0 : Choice of language is independent of gender. (A1) Notes: Do not accept “not related” or “not correlated”.
(b) 2 (A1)
(c) = 23 (M1)(A1)(G2)
Notes: Award (M1)for correct substituted formula, (A1)for 23.
(d) χ2 = 4.77 (G2)
Notes: If answer is incorrect, award (M1)for correct substitution in the correct formula (all terms).
(e) Accept H0 since
(5.99) or p-value (0.0923) > 0.05 (R1)(A1)(ft)
Notes: Do not award (R0)(A1). Follow through from their (d) and (b).
[8]
2. (a) Chosen profession is independent of gender. (A1)
OR
There is no association between gender and chosen profession. (A1) (C1)
Note: Do not accept “not related”, “not correlated” or “not influenced”. (b) 2 (A1) (C1) (c) (M1) OR × 400 (M1) = 36 (A1) (C2) (d) p-value > 0.05 (R1) Accept H0 (A1) (C2)
Note: Do not award (R0)(A1).
150 69 50 2 2 crit calc 400 80 180 400 80 400 180
3. (a) (i) Type of coffee drunk is independent of gender. (A1)
Note: Accept “not associated”. Do not accept “not related” “not correlated” or “not influenced”.
(ii) Type of coffee drunk is notindependent of gender. (A1)(ft) (C2)
Note: If hypotheses are reversed award (A0)(A1)(ft).
(b) 4 (A1) (C1) (c) χ2(crit) = 9.488 (A1)(ft) (C1) Note: Accept 9.49 (d) χ2(calc) < χ2(crit), 8.73 < 9.488 (R1) Accept the null hypothesis (Accept type of coffee is independent
of gender). (A1)(ft) (C2)
Notes: Follow through from their answer to part (c). Do not award (R0)(A1).
[6]
4. (a) H0: The (average) number of meals per day a student has and
gender are independent (A1)
Note: For “independent” accept “not associated” but do not accept “not related” or “not correlated”.
(b) 2 (A1)
(c) 5.99 (accept 5.991) (A1)(ft)
Note: Follow through from their part (b).
(d) = 12.6 = 13 or × 100 = 12.6 = 13 (M1)(A1)(AG)
Notes: Award (M1) for correct formula and (A1) for correct substitution. Unrounded answer must be seen for the (A1) to be awarded.
(e) 0.0321 (G2)
Note: For 0.032 award (G1)(G1)(AP). For 0.03 with no working award (G0).
(f) 0.0321 < 5.99 or 0.984 > 0.05 (R1)
accept H0 (A1)(ft)
Note: If reason is incorrect both marks are lost, do not award (R0)(A1).
[9] 100 45 28 100 25 100 28
5. (a) (3 – l)(4 – l) (M1)
= 6 (A1) (C2)
(b) The preferred type of drink is independent of age. (A1) (C1)
Note: For independent accept “not associated” but do not accept “not related” or “not correlated”.
(c) 12.592 (12.6) (A1) (C1)
(d) Reject null hypothesis as critical value < (A1)(R1)(ft)
Note: (ft) from their value in (c).
OR
Reject null hypothesis as p-value < 0.05 (A1)(R1) (C2)
Notes: Do not award (A1)(R0).
Award the (R1) for comparison of correct values.
[6]
6. (a) 28 (A1)
(b) (M1)(A1)(ft)
Note: Award (M1) for correct formula, (A1) for correct substitution.
= 12.6 (AG)
Note: Do not award (A1) unless 12.6 seen.
(c) (i) the favourite car colour is independent of gender. (A1)
Note: Accept there is no association between gender and favourite car colour.
Do not accept ‘not related’ or ‘not correlated’.
(ii) 2 (A1)
(iii) 5.991 (5.99) (A1)(ft)
Note: Follow through from (c)(ii) for their degrees of freedom. Note: Accept any accuracy beyond 3 s.f.
(iv) Accept the null hypothesis since 1.367 < 5.991 (A1)(ft)(R1)
Note: Allow “Do not reject”. Follow through from their null hypothesis and their critical value.
Full credit for use of p-values from GDC [p = 0.505] Do not award (A1)(R0). Award (R1) for valid comparison.
[8] 2 calc 100 100 45 100 28 100 45 28
