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

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

Write down

A customer is selected at random.

State $H_{0}$ , 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 $31$ , correct to two significant figures.

[2]

c.

the $χ_{2}$ statistic.

[2]

d.i.

the $p$ -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

$(H_{0} :)$ 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.

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

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

$30.82 (30.8)$ (A1)

$31$ (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.

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

[2 marks]

d.i.

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

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

[1 mark]

d.ii.

$0.265 > 0.10$ OR $2.66 < 4.605$ (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 $χ^{2}$ statistic to the $χ^{2}$ critical value or their $p$ -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.

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

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

[2 marks]

f.i.

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

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

[2 marks]

f.ii.

$\frac{34}{69} (0.493, 0.492753 \dots, 49.3 %)$ (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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18M.2.SL.TZ2.T_2f:
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17N.2.SL.TZ0.T_1b.ii:
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18M.2.SL.TZ2.T_2d:
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19M.1.SL.TZ2.T_6c:
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State, giving a reason, which school cafeteria has more meals that do not meet the requirement.
17N.1.SL.TZ0.T_1a.ii:
For the students in this group write down the median age.
19M.1.SL.TZ1.T_5b:
Write down the probability that a bouquet of roses sold is not small.
18M.1.SL.TZ2.T_7a:
State the number of boys who answered questions in Portuguese.
19N.2.SL.TZ0.T_1d.ii:
the $p$ -value.
17M.1.SL.TZ2.T_7c:
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18M.2.SL.TZ1.T_2d.i:
Write down the χ² statistic.
18M.2.SL.TZ2.T_2b.ii:
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19M.1.SL.TZ2.T_12a.ii:
$q$ .
18M.2.SL.TZ1.T_2a:
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17N.2.SL.TZ0.T_1c.i:
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19M.1.SL.TZ2.T_12c:
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17N.2.SL.TZ0.T_1e.ii:
Write down the number of degrees of freedom.
19N.2.SL.TZ0.T_1d.i:
the $χ_{2}$ statistic.
19N.2.SL.TZ0.T_1b:
Write down the number of degrees of freedom.
17N.2.SL.TZ0.T_1b.i:
Write down, for $N$ , the modal class;
19N.1.SL.TZ0.T_15a:
Write down the median rehearsal time.
19N.2.SL.TZ0.T_1a:
State $H_{0}$ , the null hypothesis for this test.
17N.2.SL.TZ0.T_1a:
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17N.2.SL.TZ0.T_1c.ii:
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19N.1.SL.TZ0.T_13a:
State whether the weight of the eggs is a continuous or discrete variable.
17N.2.SL.TZ0.S_8e:
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19M.1.SL.TZ1.T_5c:
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(ii) the standard deviation.
18M.1.SL.TZ2.T_12a:
Write down the mid-interval value for 10 ≤ t < 15.
18M.1.SL.TZ2.T_12b.i:
Write down the total number of customers in terms of k.
17N.2.SL.TZ0.S_8d.i:
Find the value of $k$ ;
17N.2.SL.TZ0.S_8b:
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EXN.1.SL.TZ0.4b.ii:
Find the lower quartile.
19N.2.SL.TZ0.T_1e:
State the conclusion for this test. Give a reason for your answer.
EXN.1.SL.TZ0.4b.iii:
Find the upper quartile.
19N.2.SL.TZ0.T_1c:
Show that the expected number of children who chose shrimp is $31$ , correct to two significant figures.
17N.2.SL.TZ0.T_1f.i:
Write down the $p$ -value for the test;
17N.2.SL.TZ0.T_1e.i:
Write down the null hypothesis for this test;
17N.2.SL.TZ0.S_8d.ii:
Find the number of hives that have a high production.
17N.2.SL.TZ0.T_1g:
State the result of the test. Give a reason for your answer.
18M.2.SL.TZ2.T_2c:
Hence write down the interquartile range.
17M.1.SL.TZ2.T_7b:
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18M.2.SL.TZ2.T_2b.i:
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18M.2.SL.TZ2.T_2b.iii:
Use the cumulative frequency curve to find the upper quartile.
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Calculate the probability that the customer is an adult.
20N.1.SL.TZ0.S_8a:
Find the value of $a$ .
20N.1.SL.TZ0.T_3b:
Use your graphic display calculator to find an estimate of the standard deviation of the weights of mangoes from this harvest.
20N.1.SL.TZ0.T_3c:
On the grid below, draw a histogram for the data in the table.
17N.2.SL.TZ0.T_1d:
Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.
20N.1.SL.TZ0.S_8b:
Write down the value of the median distance in kilometres (km).
20N.1.SL.TZ0.S_8d:
Find $m$ .
16N.1.SL.TZ0.T_2b:
Draw a box-and-whisker diagram on the grid below to represent the Vitamin C content, in milligrams, for this sample.
17M.1.SL.TZ1.T_1a:
Complete the following table.
EXN.1.SL.TZ0.4a:
Find the number of potatoes in the sample with a weight of more than $200$ grams.
EXN.1.SL.TZ0.4c:
The weight of the smallest potato in the sample is $20$ grams and the weight of the largest is $400$ grams.

Use the scale shown below to draw a box and whisker diagram showing the distribution of the weights of the potatoes. You may assume there are no outliers.
16N.2.SL.TZ0.S_8a:
Find the mean number of hours spent browsing the Internet.
18M.2.SL.TZ1.T_2e:
State, with a reason, whether you would reject the null hypothesis.
19N.1.SL.TZ0.T_15b:
State whether Stephen is correct. Give a reason for your answer.
20N.1.SL.TZ0.S_8c:
Find the value of $b$ .
20N.1.SL.TZ0.T_3a:
Write down the modal group for these data.
20N.1.SL.TZ0.S_8e:
The first $150$ athletes that completed the race won a prize.

Given that an athlete took between $22$ and $m$ minutes to complete the $5 km$ race, calculate the probability that they won a prize.
18M.1.SL.TZ2.T_7c:
Two girls are selected at random.

Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.
18M.2.SL.TZ1.T_2b:
Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.
21M.2.SL.TZ1.1b:
Write down the median time to read the book.
16N.2.SL.TZ0.S_8d:
(i) Find the number of students who spent between 25 and 30 hours browsing the Internet.

(ii) Given that 10% of the students spent more than k hours browsing the Internet, find the maximum value of $k$ .
18M.1.SL.TZ2.T_7b:
Find the probability that the boy answered questions in Hindi.
21M.1.SL.TZ2.5a.ii:
the lower quartile.
21M.1.SL.TZ2.5a.i:
the minimum number of sick days taken during the year.
18M.2.SL.TZ1.T_2f:
Write down the probability that this flight arrived on time.
18M.2.SL.TZ1.T_2h:
Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.
19N.1.SL.TZ0.T_15c:
On $k$ days, Stephen practiced exactly $24$ minutes.

Find the possible values of $k$ .
21M.2.SL.TZ1.1c:
Calculate the interquartile range.
19N.1.SL.TZ0.T_13d:
The mean weight of these eggs is 64.9 grams, correct to three significant figures.

Use the table and your answer to part (c) to find the smallest possible number of eggs that could be within one standard deviation of the mean.
16N.2.SL.TZ0.S_8c:
During week 3 each student spent 5% less time browsing the Internet than during week 1. For week 3, find

(i) the median;

(ii) the variance.
21M.1.SL.TZ2.5a.iii:
the median.
21M.1.SL.TZ2.5b:
Paul claims that this box and whisker diagram can be used to infer that the percentage of employees who took fewer than six sick days is smaller than the percentage of employees who took more than eleven sick days.

State whether Paul is correct. Justify your answer.
21N.1.SL.TZ0.9c.i:
Find the number of students who obtained a grade $5$ .
21N.1.SL.TZ0.9a:
Find the median of the scores obtained.
21N.1.SL.TZ0.9b:
Find an expression for $a$ in terms of $b$ .
21N.1.SL.TZ0.9c.ii:
Find the minimum score needed to obtain a grade $5$ .

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