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Date	November 2017	Marks available	1	Reference code	17N.2.SL.TZ0.T_1
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Estimate	Question number	T_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]

a.

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]

d.

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]

g.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

discrete (A1)

[1 mark]

a.

$11 \leqslant N \leqslant 20$ (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]

d.

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)

OR

(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]

g.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

d.

[N/A]

e.i.

[N/A]

e.ii.

[N/A]

f.i.

[N/A]

f.ii.

[N/A]

g.

Syllabus sections

Topic 4—Statistics and probability » SL 4.2—Histograms, CF graphs, box plots

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Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.
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Write down the null hypothesis for this test;
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Write down the number of degrees of freedom.
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Find the interquartile range.
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Write down the modal class.
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Write down the mid-interval value of the modal class.
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Use the cumulative frequency graph to estimate the number of students whose final examination result was below this grade boundary.
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Hide related questions

Topic 4—Statistics and probability » SL 4.3—Mean, median, mode. Mean of grouped data, standard deviation. Quartiles, IQR

Topic 4—Statistics and probability