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Date	November 2019	Marks available	2	Reference code	19N.2.SL.TZ0.T_1
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Calculate	Question number	T_1	Adapted from	N/A

Question

Casanova restaurant offers a set menu where a customer chooses one of the following meals: pasta, fish or shrimp.

The manager surveyed $150$ $150$ customers and recorded the customer’s age and chosen meal. The data is shown in the following table.

A $χ^{2}$ $χ^{2}$ test was performed at the $10 %$ $10 %$ significance level. The critical value for this test is $4.605$ $4.605$ .

Write down

A customer is selected at random.

State $H_{0}$ $H_{0}$ , the null hypothesis for this test.

[1]

a.

Write down the number of degrees of freedom.

[1]

b.

Show that the expected number of children who chose shrimp is $31$ $31$ , correct to two significant figures.

[2]

c.

the $χ_{2}$ $χ_{2}$ statistic.

[2]

d.i.

the $p$ $p$ -value.

[1]

d.ii.

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

[2]

e.

Calculate the probability that the customer is an adult.

[2]

f.i.

Calculate the probability that the customer is an adult or that the customer chose shrimp.

[2]

f.ii.

Given that the customer is a child, calculate the probability that they chose pasta or fish.

[2]

f.iii.

Markscheme

$(H_{0} :)$ $(H_{0} :)$ choice of meal is independent of age (or equivalent) (A1)

Note: Accept "not associated" or "not dependent" instead of independent. In lieu of "age", accept an equivalent alternative such as "being a child or adult".

[1 mark]

a.

2 (A1)

[1 mark]

b.

$\frac{69}{150} \times \frac{67}{150} \times 150$ $\frac{69}{150} \times \frac{67}{150} \times 150$ OR $\frac{69 \times 67}{150}$ $\frac{69 \times 67}{150}$ (M1)

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

$30.82 (30.8)$ $30.82 (30.8)$ (A1)

$31$ $31$ (AG)

Note: Both an unrounded answer that rounds to the given answer and rounded answer must be seen for the (A1) to be awarded.

[2 marks]

c.

$(χ_{calc}^{2} =) 2.66 (2.657537 \dots)$ $(χ_{calc}^{2} =) 2.66 (2.657537 \dots)$ (G2)

[2 marks]

d.i.

( $p$ $p$ -value $=$ $=$ ) $0.265 (0.264803 \dots)$ $0.265 (0.264803 \dots)$ (G1)

Note: Award (G0)(G2) if the $χ^{2}$ $χ^{2}$ statistic is missing or incorrect and the $p$ $p$ -value is correct.

[1 mark]

d.ii.

$0.265 > 0.10$ $0.265 > 0.10$ OR $2.66 < 4.605$ $2.66 < 4.605$ (R1)(ft)

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

OR

the choice of meal is independent of age (or equivalent) (A1)(ft)

Note: Award (R1)(ft)) for a correct comparison of either their $χ^{2}$ $χ^{2}$ statistic to the $χ^{2}$ $χ^{2}$ critical value or their $p$ $p$ -value to the significance level.
Condone “accept” in place of “not reject”.
Follow through from parts (a) and (d).

Do not award (A1)(ft)(R0).

[2 marks]

e.

$\frac{81}{150} (\frac{27}{50}, 0.54, 54 %)$ $\frac{81}{150} (\frac{27}{50}, 0.54, 54 %)$ (A1)(A1)(G2)

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

[2 marks]

f.i.

$\frac{116}{150} (\frac{58}{75}, 0.773, 0.773333 \dots, 77.3 %)$ $\frac{116}{150} (\frac{58}{75}, 0.773, 0.773333 \dots, 77.3 %)$ (A1)(A1)(G2)

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

[2 marks]

f.ii.

$\frac{34}{69} (0.493, 0.492753 \dots, 49.3 %)$ $\frac{34}{69} (0.493, 0.492753 \dots, 49.3 %)$ (A1)(A1)(G2)

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

[2 marks]

f.iii.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

e.

[N/A]

f.i.

[N/A]

f.ii.

[N/A]

f.iii.

Syllabus sections

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

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19N.1.SL.TZ0.T_15b:
State whether Stephen is correct. Give a reason for your answer.
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21M.1.SL.TZ2.5a.ii:
the lower quartile.
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