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Date	November 2016	Marks available	6	Reference code	16N.2.SL.TZ0.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_8	Adapted from	N/A

Question

Ten students were surveyed about the number of hours, $x$ , they spent browsing the Internet during week 1 of the school year. The results of the survey are given below.

$\sum\limits_{i = 1}^{10} {{x_i} = 252,{\text{ }}\sigma = 5{\text{ and median}} = 27.}$

During week 4, the survey was extended to all 200 students in the school. The results are shown in the cumulative frequency graph:

N16/5/MATME/SP2/ENG/TZ0/08.d

Find the mean number of hours spent browsing the Internet.

[2]

During week 2, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week 2, write down

(i) the mean;

(ii) the standard deviation.

[2]

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.

[6]

(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 $k$ .

[6]

Markscheme

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

attempt to substitute into formula for mean (M1)

eg $\,\,\,\,\,$ $\frac{{\Sigma x}}{{10}},{\text{ }}\frac{{252}}{n},{\text{ }}\frac{{252}}{{10}}$

mean $= 25.2{\text{ (hours)}}$ A1 N2

[2 marks]

(i) mean $= 30.2{\text{ (hours)}}$ A1 N1

(ii) $\sigma = 5{\text{ (hours)}}$ A1 N1

[2 marks]

(i) valid approach (M1)

eg $\,\,\,\,\,$ 95%, 5% of 27

correct working (A1)

eg $\,\,\,\,\,$ $0.95 \times 27,{\text{ }}27 - (5\% {\text{ of }}27)$

median $= 25.65{\text{ (exact), }}25.7{\text{ (hours)}}$ A1 N2

(ii) METHOD 1

variance $= {({\text{standard deviation}})^2}$ (seen anywhere) (A1)

valid attempt to find new standard deviation (M1)

eg $\,\,\,\,\,$ ${\sigma _{new}} = 0.95 \times 5,{\text{ }}4.75$

variance $= 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6$ A1 N2

METHOD 2

variance $= {({\text{standard deviation}})^2}$ (seen anywhere) (A1)

valid attempt to find new variance (M1)

eg $\,\,\,\,\,$ ${0.95^2}{\text{ }},{\text{ }}0.9025 \times {\sigma ^2}$

new variance $= 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6$ A1 N2

[6 marks]

(i) both correct frequencies (A1)

eg $\,\,\,\,\,$ 80, 150

subtracting their frequencies in either order (M1)

eg $\,\,\,\,\,$ $150 - 80,{\text{ }}80 - 150$

70 (students) A1 N2

(ii) evidence of a valid approach (M1)

eg $\,\,\,\,\,$ 10% of 200, 90%

correct working (A1)

eg $\,\,\,\,\,$ $0.90 \times 200,{\text{ }}200 - 20$ , 180 students

$k = 35$ A1 N3

[6 marks]

Examiners report

[N/A]

Syllabus sections

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

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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

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Markscheme

Examiners report

Syllabus sections

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