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Date May 2017 Marks available 2 Reference code 17M.2.SL.TZ2.T_1
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Give a reason and State Question number T_1 Adapted from N/A

Question

In a school, all Mathematical Studies SL students were given a test. The test contained four questions, each one on a different topic from the syllabus. The quality of each response was classified as satisfactory or not satisfactory. Each student answered only three of the four questions, each on a separate answer sheet.

The table below shows the number of satisfactory and not satisfactory responses for each question.

M17/5/MATSD/SP2/ENG/TZ2/01

A χ 2 test is carried out at the 5% significance level for the data in the table.

The critical value for this test is 7.815.

If the teacher chooses a response at random, find the probability that it is a response to the Calculus question;

[2]
a.i.

If the teacher chooses a response at random, find the probability that it is a satisfactory response to the Calculus question;

[2]
a.ii.

If the teacher chooses a response at random, find the probability that it is a satisfactory response, given that it is a response to the Calculus question.

[2]
a.iii.

The teacher groups the responses by topic, and chooses two responses to the Logic question. Find the probability that both are not satisfactory.

[3]
b.

State the null hypothesis for this test.

[1]
c.

Show that the expected frequency of satisfactory Calculus responses is 12.

[1]
d.

Write down the number of degrees of freedom for this test.

[1]
e.

Use your graphic display calculator to find the χ 2 statistic for this data.

[2]
f.

State the conclusion of this χ 2 test. Give a reason for your answer.

[2]
g.

Markscheme

1 5   ( 18 90 ;   0.2 ;   20 % )     (A1)(A1)(G2)

 

Note:     Award (A1) for correct numerator, (A1) for correct denominator.

 

[2 marks]

a.i.

1 9   ( 10 90 ;   0 . 1 ¯ ;   0.111111 ;   11.1 % )     (A1)(A1)(G2)

 

Note:     Award (A1) for correct numerator, (A1) for correct denominator.

 

[2 marks]

a.ii.

5 9   ( 10 18 ;   0. 5 ¯ ;   0.555556 ;   55.6 % )     (A1)(A1)(G2)

 

Note:     Award (A1) for correct numerator, (A1) for correct denominator.

 

[2 marks]

a.iii.

6 20 × 5 19     (A1)(M1)

 

Note:     Award (A1) for two correct fractions seen, (M1) for multiplying their two fractions.

 

3 38   ( 30 380 ;   0.0789473 ;   7.89 % )     (A1)(G2)

[3 marks]

b.

H 0 : quality (of response) and topic (from the syllabus) are independent     (A1)

 

Note:     Accept there is no association between quality (of response) and topic (from the syllabus). Do not accept “not related” or “not correlated” or “influenced”.

 

[1 mark]

c.

18 90 × 60 90 × 90 OR 18 × 60 90     (M1)

 

Note:     Award (M1) for correct substitution in expected value formula.

 

( = )   12     (AG)

 

Note:     The conclusion, ( = )   12 , must be seen for the (A1) to be awarded.

 

[1 mark]

d.

3     (A1)

[1 mark]

e.

( χ c a l c 2 = )   1.46   ( 1.46 36 ¯ ;   1.46363 )     (G2)

[2 marks]

f.

1.46 < 7.815 OR 0.690688 > 0.05     (R1)

the null hypothesis is not rejected     (A1)(ft)

OR

the quality of the response and the topic are independent     (A1)(ft)

 

Note:     Award (R1) for a correct comparison of either their χ 2 statistic to the χ 2 critical value or the correct p -value 0.690688… to the test level, award (A1)(ft) for the correct result from that comparison. Accept “ χ calc 2 < χ crit 2 ” for the comparison, but only if their χ calc 2 value is explicitly seen in part (f). Follow through from their answers to part (f) and part (c). Do not award (R0)(A1).

 

[2 marks]

g.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
a.iii.
[N/A]
b.
[N/A]
c.
[N/A]
d.
[N/A]
e.
[N/A]
f.
[N/A]
g.

Syllabus sections

Topic 4—Statistics and probability » SL 4.11—Expected, observed, hypotheses, chi squared, gof, t-test
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Topic 4—Statistics and probability

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