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Date	November 2017	Marks available	1	Reference code	17N.2.sl.TZ0.1
Level	SL only	Paper	2	Time zone	TZ0
Command term	State	Question number	1	Adapted from	N/A

Question

A group of 800 students answered 40 questions on a category of their choice out of History, Science and Literature.

For each student the category and the number of correct answers, \(N\), was recorded. The results obtained are represented in the following table.

N17/5/MATSD/SP2/ENG/TZ0/01

A \({\chi ^2}\) test at the 5% significance level is carried out on the results. The critical value for this test is 12.592.

State whether \(N\) is a discrete or a continuous variable.

[1]

Write down, for \(N\), the modal class;

[1]

b.i.

Write down, for \(N\), the mid-interval value of the modal class.

[1]

b.ii.

Use your graphic display calculator to estimate the mean of \(N\);

[2]

c.i.

Use your graphic display calculator to estimate the standard deviation of \(N\).

[1]

c.ii.

Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.

[2]

Write down the null hypothesis for this test;

[1]

e.i.

Write down the number of degrees of freedom.

[1]

e.ii.

Write down the \(p\)-value for the test;

[1]

f.i.

Write down the \({\chi ^2}\) statistic.

[2]

f.ii.

State the result of the test. Give a reason for your answer.

[2]

Markscheme

discrete (A1)

[1 mark]

\(11 \leqslant N \leqslant 20\) (A1)

[1 mark]

b.i.

15.5 (A1)(ft)

Note: Follow through from part (b)(i).

[1 mark]

b.ii.

\(21.2{\text{ }}(21.2125)\) (G2)

[2 marks]

c.i.

\(9.60{\text{ }}(9.60428 \ldots )\) (G1)

[1 marks]

c.ii.

\(\frac{{260}}{{800}} \times \frac{{157}}{{800}} \times 800\)\(\,\,\,\)OR\(\,\,\,\)\(\frac{{260 \times 157}}{{800}}\) (M1)

Note: Award (M1) for correct substitution into expected frequency formula.

\( = 51.0{\text{ }}(51.025)\) (A1)(G2)

[2 marks]

choice of category and number of correct answers are independent (A1)

Notes: Accept “no association” between (choice of) category and number of correct answers. Do not accept “not related” or “not correlated” or “influenced”.

[1 mark]

e.i.

6 (A1)

[1 mark]

e.ii.

\(0.0644{\text{ }}(0.0644123 \ldots )\) (G1)

[1 mark]

f.i.

\(11.9{\text{ }}(11.8924 \ldots )\) (G2)

[2 marks]

f.ii.

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

(choice of) category and number of correct answers are independent (A1)(ft)

as \(11.9 < 12.592\)\(\,\,\,\)OR\(\,\,\,\)\(0.0644 > 0.05\) (R1)

Notes: Award (R1) for a correct comparison of either their \({\chi ^2}\) statistic to the \({\chi ^2}\) critical value or their \(p\)-value to the significance level. Award (A1)(ft) from that comparison.

Follow through from part (f). Do not award (A1)(ft)(R0).

[2 marks]

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

e.i.

[N/A]

e.ii.

[N/A]

f.i.

[N/A]

f.ii.

[N/A]

Syllabus sections

Topic 4 - Statistical applications » 4.4 » The \({\chi ^2}\) test for independence: formulation of null and alternative hypotheses; significance levels; contingency tables; expected frequencies; degrees of freedom; \(p\)-values.

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